math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.5s

Alternatives: 29

Speedup: 1.5×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

      \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

   5. +-commutative N/A

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

   6. sub0-neg N/A

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

   7. cosh-undef N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. exp-0 N/A

       \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

       \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

   12. exp-0 N/A

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

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

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

   14. sin-lowering-sin.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \cosh im \cdot \sin re \]
6. Add Preprocessing

Alternative 2: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot 0.5\\ t_1 := t\_0 \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) 0.5)) (t_1 (* t_0 (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (* (cosh im) (* re (fma re (* re -0.16666666666666666) 1.0)))
     (if (<= t_1 1.0) (* t_0 (fma im im 2.0)) (* (cosh im) re)))))

C

double code(double re, double im) {
	double t_0 = sin(re) * 0.5;
	double t_1 = t_0 * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(im) * (re * fma(re, (re * -0.16666666666666666), 1.0));
	} else if (t_1 <= 1.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = cosh(im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * 0.5)
	t_1 = Float64(t_0 * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(im) * Float64(re * fma(re, Float64(re * -0.16666666666666666), 1.0)));
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(cosh(im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 68.4

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

   7. Simplified 68.4%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 98.8

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

   5. Simplified 98.8%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.9%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 87.9

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

   9. Applied egg-rr 87.9%

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

4. Final simplification 89.2%

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

Alternative 3: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot 0.5\\ t_1 := t\_0 \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) 0.5)) (t_1 (* t_0 (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma (* im im) 0.001388888888888889 0.041666666666666664)
       (* im (* im (* im im)))
       (fma 0.5 (* im im) 1.0))
      (*
       re
       (fma
        (* re re)
        (fma
         (* re re)
         (fma re (* re -0.0001984126984126984) 0.008333333333333333)
         -0.16666666666666666)
        1.0)))
     (if (<= t_1 1.0) (* t_0 (fma im im 2.0)) (* (cosh im) re)))))

C

double code(double re, double im) {
	double t_0 = sin(re) * 0.5;
	double t_1 = t_0 * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma((im * im), 0.001388888888888889, 0.041666666666666664), (im * (im * (im * im))), fma(0.5, (im * im), 1.0)) * (re * fma((re * re), fma((re * re), fma(re, (re * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), 1.0));
	} else if (t_1 <= 1.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = cosh(im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * 0.5)
	t_1 = Float64(t_0 * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), Float64(im * Float64(im * Float64(im * im))), fma(0.5, Float64(im * im), 1.0)) * Float64(re * fma(Float64(re * re), fma(Float64(re * re), fma(re, Float64(re * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), 1.0)));
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(cosh(im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 83.4%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \color{blue}{\frac{-1}{6}}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}, \frac{-1}{6}\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{5040} \cdot {re}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{5040} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \frac{-1}{5040}\right)} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      16. *-lowering-*.f64 58.6

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot -0.0001984126984126984}, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   8. Simplified 58.6%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 98.8

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

   5. Simplified 98.8%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.9%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 87.9

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

   9. Applied egg-rr 87.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 1:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma (* im im) 0.001388888888888889 0.041666666666666664)
       (* im (* im (* im im)))
       (fma 0.5 (* im im) 1.0))
      (*
       re
       (fma
        (* re re)
        (fma
         (* re re)
         (fma re (* re -0.0001984126984126984) 0.008333333333333333)
         -0.16666666666666666)
        1.0)))
     (if (<= t_0 1.0) (sin re) (* (cosh im) re)))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma((im * im), 0.001388888888888889, 0.041666666666666664), (im * (im * (im * im))), fma(0.5, (im * im), 1.0)) * (re * fma((re * re), fma((re * re), fma(re, (re * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), 1.0));
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = cosh(im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), Float64(im * Float64(im * Float64(im * im))), fma(0.5, Float64(im * im), 1.0)) * Float64(re * fma(Float64(re * re), fma(Float64(re * re), fma(re, Float64(re * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), 1.0)));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(cosh(im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 83.4%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \color{blue}{\frac{-1}{6}}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}, \frac{-1}{6}\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{5040} \cdot {re}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{5040} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \frac{-1}{5040}\right)} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      16. *-lowering-*.f64 58.6

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot -0.0001984126984126984}, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   8. Simplified 58.6%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 98.1

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

   5. Simplified 98.1%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.9%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 87.9

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

   9. Applied egg-rr 87.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\\ t_1 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ t_2 := im \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_2, \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(t\_0, t\_2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) 0.001388888888888889 0.041666666666666664))
        (t_1 (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))))
        (t_2 (* im (* im (* im im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma t_0 t_2 (fma 0.5 (* im im) 1.0))
      (*
       re
       (fma
        (* re re)
        (fma
         (* re re)
         (fma re (* re -0.0001984126984126984) 0.008333333333333333)
         -0.16666666666666666)
        1.0)))
     (if (<= t_1 1.0)
       (sin re)
       (*
        (*
         re
         (fma
          (* re re)
          (fma (* re re) 0.008333333333333333 -0.16666666666666666)
          1.0))
        (fma t_0 t_2 1.0))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), 0.001388888888888889, 0.041666666666666664);
	double t_1 = (sin(re) * 0.5) * (exp((0.0 - im)) + exp(im));
	double t_2 = im * (im * (im * im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(t_0, t_2, fma(0.5, (im * im), 1.0)) * (re * fma((re * re), fma((re * re), fma(re, (re * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), 1.0));
	} else if (t_1 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = (re * fma((re * re), fma((re * re), 0.008333333333333333, -0.16666666666666666), 1.0)) * fma(t_0, t_2, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)
	t_1 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	t_2 = Float64(im * Float64(im * Float64(im * im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(t_0, t_2, fma(0.5, Float64(im * im), 1.0)) * Float64(re * fma(Float64(re * re), fma(Float64(re * re), fma(re, Float64(re * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), 1.0)));
	elseif (t_1 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666), 1.0)) * fma(t_0, t_2, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(t$95$0 * t$95$2 + N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[re], $MachinePrecision], N[(N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 83.4%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) + \color{blue}{\frac{-1}{6}}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}, \frac{-1}{6}\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{5040} \cdot {re}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{5040} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \frac{-1}{5040}\right)} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      16. *-lowering-*.f64 58.6

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot -0.0001984126984126984}, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   8. Simplified 58.6%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 98.1

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

   5. Simplified 98.1%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 88.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      11. *-lowering-*.f64 80.5

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   8. Simplified 80.5%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{1}\right) \]
   10. Step-by-step derivation

   11. Simplified 80.5%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{1}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 1.0)
   (*
    (sin re)
    (fma
     (fma (* im im) 0.001388888888888889 0.041666666666666664)
     (* im (* im (* im im)))
     (fma 0.5 (* im im) 1.0)))
   (* (cosh im) re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 1.0) {
		tmp = sin(re) * fma(fma((im * im), 0.001388888888888889, 0.041666666666666664), (im * (im * (im * im))), fma(0.5, (im * im), 1.0));
	} else {
		tmp = cosh(im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 1.0)
		tmp = Float64(sin(re) * fma(fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), Float64(im * Float64(im * Float64(im * im))), fma(0.5, Float64(im * im), 1.0)));
	else
		tmp = Float64(cosh(im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 94.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.9%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 87.9

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

   9. Applied egg-rr 87.9%

      \[\leadsto \color{blue}{\cosh im} \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot 0.5\\ \mathbf{if}\;t\_0 \cdot \left(e^{0 - im} + e^{im}\right) \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), 1\right), 0\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) 0.5)))
   (if (<= (* t_0 (+ (exp (- 0.0 im)) (exp im))) 1.0)
     (*
      t_0
      (fma
       im
       (fma
        im
        (fma
         im
         (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
         1.0)
        0.0)
       2.0))
     (* (cosh im) re))))

C

double code(double re, double im) {
	double t_0 = sin(re) * 0.5;
	double tmp;
	if ((t_0 * (exp((0.0 - im)) + exp(im))) <= 1.0) {
		tmp = t_0 * fma(im, fma(im, fma(im, (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), 1.0), 0.0), 2.0);
	} else {
		tmp = cosh(im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * 0.5)
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 1.0)
		tmp = Float64(t_0 * fma(im, fma(im, fma(im, Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), 1.0), 0.0), 2.0));
	else
		tmp = Float64(cosh(im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(t$95$0 * N[(im * N[(im * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 0.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

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

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Simplified 94.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), 1\right), 0\right), 2\right)} \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.9%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 87.9

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

   9. Applied egg-rr 87.9%

      \[\leadsto \color{blue}{\cosh im} \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 1:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), 1\right), 0\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 1.0)
   (* (sin re) (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
   (* (cosh im) re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 1.0) {
		tmp = sin(re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = cosh(im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 1.0)
		tmp = Float64(sin(re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(cosh(im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 93.1%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Simplified 87.9%

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

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 87.9

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

   9. Applied egg-rr 87.9%

      \[\leadsto \color{blue}{\cosh im} \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\\ \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 2 \cdot 10^{-141}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, t\_0, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(t\_0, im \cdot \left(im \cdot \left(im \cdot im\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) 0.001388888888888889 0.041666666666666664)))
   (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 2e-141)
     (*
      re
      (*
       (fma re (* re -0.16666666666666666) 1.0)
       (fma (* im im) (fma (* im im) t_0 0.5) 1.0)))
     (*
      (*
       re
       (fma
        (* re re)
        (fma (* re re) 0.008333333333333333 -0.16666666666666666)
        1.0))
      (fma t_0 (* im (* im (* im im))) 1.0)))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), 0.001388888888888889, 0.041666666666666664);
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 2e-141) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma((im * im), fma((im * im), t_0, 0.5), 1.0));
	} else {
		tmp = (re * fma((re * re), fma((re * re), 0.008333333333333333, -0.16666666666666666), 1.0)) * fma(t_0, (im * (im * (im * im))), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 2e-141)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(im * im), fma(Float64(im * im), t_0, 0.5), 1.0)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666), 1.0)) * fma(t_0, Float64(im * Float64(im * Float64(im * im))), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-141], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * t$95$0 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-141

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 91.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 62.5%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)} \]

   if 2.0000000000000001e-141 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 93.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      11. *-lowering-*.f64 60.3

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   8. Simplified 60.3%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{1}\right) \]
   10. Step-by-step derivation

   11. Simplified 60.3%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 2 \cdot 10^{-141}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\\ \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, t\_0, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(t\_0, im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) 0.001388888888888889 0.041666666666666664)))
   (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
     (*
      re
      (*
       (fma re (* re -0.16666666666666666) 1.0)
       (fma (* im im) (fma (* im im) t_0 0.5) 1.0)))
     (* re (fma t_0 (* im (* im (* im im))) (fma 0.5 (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), 0.001388888888888889, 0.041666666666666664);
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma((im * im), fma((im * im), t_0, 0.5), 1.0));
	} else {
		tmp = re * fma(t_0, (im * (im * (im * im))), fma(0.5, (im * im), 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(im * im), fma(Float64(im * im), t_0, 0.5), 1.0)));
	else
		tmp = Float64(re * fma(t_0, Float64(im * Float64(im * Float64(im * im))), fma(0.5, Float64(im * im), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * t$95$0 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(t$95$0 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 92.9%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 67.7%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)} \]

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 51.5%

      \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (*
    (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0)
    (fma (* re (* re re)) -0.16666666666666666 re))
   (*
    re
    (fma
     (fma (* im im) 0.001388888888888889 0.041666666666666664)
     (* im (* im (* im im)))
     (fma 0.5 (* im im) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0) * fma((re * (re * re)), -0.16666666666666666, re);
	} else {
		tmp = re * fma(fma((im * im), 0.001388888888888889, 0.041666666666666664), (im * (im * (im * im))), fma(0.5, (im * im), 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = Float64(fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0) * fma(Float64(re * Float64(re * re)), -0.16666666666666666, re));
	else
		tmp = Float64(re * fma(fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), Float64(im * Float64(im * Float64(im * im))), fma(0.5, Float64(im * im), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + re), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 66.9

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \frac{-1}{6}} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \left({re}^{3} \cdot \frac{-1}{6} + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{3}, \frac{-1}{6}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot re\right)}, \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot re\right)}, \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      9. *-lowering-*.f64 66.9

         \[\leadsto \mathsf{fma}\left(re \cdot \color{blue}{\left(re \cdot re\right)}, -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   10. Applied egg-rr 66.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 51.5%

      \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (*
    (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0)
    (fma (* re (* re re)) -0.16666666666666666 re))
   (*
    re
    (fma
     (* im im)
     (fma
      (* im im)
      (fma (* im im) 0.001388888888888889 0.041666666666666664)
      0.5)
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0) * fma((re * (re * re)), -0.16666666666666666, re);
	} else {
		tmp = re * fma((im * im), fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = Float64(fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0) * fma(Float64(re * Float64(re * re)), -0.16666666666666666, re));
	else
		tmp = Float64(re * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + re), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 66.9

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \frac{-1}{6}} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \left({re}^{3} \cdot \frac{-1}{6} + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{3}, \frac{-1}{6}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot re\right)}, \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot re\right)}, \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      9. *-lowering-*.f64 66.9

         \[\leadsto \mathsf{fma}\left(re \cdot \color{blue}{\left(re \cdot re\right)}, -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   10. Applied egg-rr 66.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + 1\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + 1\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + 1\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} + 1\right)\right) \]

      7. *-commutative N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}\right) + 1\right)} \]

      9. distribute-rgt-in N/A

         \[\leadsto re \cdot \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} + 1\right) \]

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

          \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), 1\right)} \]

   8. Simplified 51.5%

      \[\leadsto \color{blue}{re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (*
    re
    (*
     (fma re (* re -0.16666666666666666) 1.0)
     (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0)))
   (*
    re
    (fma
     (* im im)
     (fma
      (* im im)
      (fma (* im im) 0.001388888888888889 0.041666666666666664)
      0.5)
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0));
	} else {
		tmp = re * fma((im * im), fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0)));
	else
		tmp = Float64(re * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

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

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\right)} \]

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + 1\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + 1\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + 1\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} + 1\right)\right) \]

      7. *-commutative N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}\right) + 1\right)} \]

      9. distribute-rgt-in N/A

         \[\leadsto re \cdot \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} + 1\right) \]

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

          \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), 1\right)} \]

   8. Simplified 51.5%

      \[\leadsto \color{blue}{re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (* (fma (* re (* re re)) -0.16666666666666666 re) (fma (* im im) 0.5 1.0))
   (*
    re
    (fma
     (* im im)
     (fma
      (* im im)
      (fma (* im im) 0.001388888888888889 0.041666666666666664)
      0.5)
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = fma((re * (re * re)), -0.16666666666666666, re) * fma((im * im), 0.5, 1.0);
	} else {
		tmp = re * fma((im * im), fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = Float64(fma(Float64(re * Float64(re * re)), -0.16666666666666666, re) * fma(Float64(im * im), 0.5, 1.0));
	else
		tmp = Float64(re * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 66.9

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \frac{-1}{6}} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \left({re}^{3} \cdot \frac{-1}{6} + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{3}, \frac{-1}{6}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot re\right)}, \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot re\right)}, \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      9. *-lowering-*.f64 66.9

         \[\leadsto \mathsf{fma}\left(re \cdot \color{blue}{\left(re \cdot re\right)}, -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   10. Applied egg-rr 66.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   11. Taylor expanded in im around 0

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

   13. Simplified 61.2%

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

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + 1\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + 1\right)\right) \]

      5. pow-sqr N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + 1\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} + 1\right)\right) \]

      7. *-commutative N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}\right) + 1\right)} \]

      9. distribute-rgt-in N/A

         \[\leadsto re \cdot \left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)} + 1\right) \]

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

          \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), 1\right)} \]

   8. Simplified 51.5%

      \[\leadsto \color{blue}{re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), 0.041666666666666664, \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (* (fma (* re (* re re)) -0.16666666666666666 re) (fma (* im im) 0.5 1.0))
   (*
    re
    (fma
     (* im (* im (* im im)))
     0.041666666666666664
     (fma 0.5 (* im im) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = fma((re * (re * re)), -0.16666666666666666, re) * fma((im * im), 0.5, 1.0);
	} else {
		tmp = re * fma((im * (im * (im * im))), 0.041666666666666664, fma(0.5, (im * im), 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = Float64(fma(Float64(re * Float64(re * re)), -0.16666666666666666, re) * fma(Float64(im * im), 0.5, 1.0));
	else
		tmp = Float64(re * fma(Float64(im * Float64(im * Float64(im * im))), 0.041666666666666664, fma(0.5, Float64(im * im), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664 + N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 66.9

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \frac{-1}{6}} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \left({re}^{3} \cdot \frac{-1}{6} + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{3}, \frac{-1}{6}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot re\right)}, \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot re\right)}, \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      9. *-lowering-*.f64 66.9

         \[\leadsto \mathsf{fma}\left(re \cdot \color{blue}{\left(re \cdot re\right)}, -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   10. Applied egg-rr 66.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   11. Taylor expanded in im around 0

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

   13. Simplified 61.2%

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

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 87.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

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

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot re \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right)} \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, 1\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot re \]

      11. *-lowering-*.f64 48.7

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right), 1\right) \cdot re \]

   8. Simplified 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot re} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \left(\color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24}\right) + \left(im \cdot im\right) \cdot \frac{1}{2}\right)} + 1\right) \cdot re \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \frac{1}{24}\right)\right)} + \left(im \cdot im\right) \cdot \frac{1}{2}\right) + 1\right) \cdot re \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)}\right) + 1\right) \cdot re \]

      4. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right)\right) + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right)} \cdot re \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \frac{1}{24}\right)} + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \cdot re \]

      6. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{24}} + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \cdot re \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)} \cdot \frac{1}{24} + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \cdot re \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), \frac{1}{24}, \frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)} \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot \left(im \cdot im\right)\right)}, \frac{1}{24}, \frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot re \]

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

          \[\leadsto \mathsf{fma}\left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}, \frac{1}{24}, \frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot re \]

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

          \[\leadsto \mathsf{fma}\left(im \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right), \frac{1}{24}, \frac{1}{2} \cdot \left(im \cdot im\right) + 1\right) \cdot re \]

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

          \[\leadsto \mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), \frac{1}{24}, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)}\right) \cdot re \]

      13. *-lowering-*.f64 49.7

          \[\leadsto \mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), 0.041666666666666664, \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right)\right) \cdot re \]

   10. Applied egg-rr 49.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), 0.041666666666666664, \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), 0.041666666666666664, \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (* (fma (* re (* re re)) -0.16666666666666666 re) (fma (* im im) 0.5 1.0))
   (* (* (* im im) (* im im)) (* re 0.041666666666666664))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = fma((re * (re * re)), -0.16666666666666666, re) * fma((im * im), 0.5, 1.0);
	} else {
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = Float64(fma(Float64(re * Float64(re * re)), -0.16666666666666666, re) * fma(Float64(im * im), 0.5, 1.0));
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(re * 0.041666666666666664));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 66.9

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot \left(re \cdot \frac{-1}{6}\right)\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \frac{-1}{6}} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. cube-unmult N/A

         \[\leadsto \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re \cdot 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \left({re}^{3} \cdot \frac{-1}{6} + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{3}, \frac{-1}{6}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. cube-unmult N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot re\right)}, \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot re\right)}, \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      9. *-lowering-*.f64 66.9

         \[\leadsto \mathsf{fma}\left(re \cdot \color{blue}{\left(re \cdot re\right)}, -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   10. Applied egg-rr 66.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   11. Taylor expanded in im around 0

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

   13. Simplified 61.2%

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

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 87.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

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

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot re \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right)} \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, 1\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot re \]

      11. *-lowering-*.f64 48.7

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right), 1\right) \cdot re \]

   8. Simplified 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot re} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right)} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       9. associate-+r+ N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot re + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right)} \]

       10. *-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(\color{blue}{re \cdot \frac{1}{24}} + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       11. associate-*r/ N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \color{blue}{\frac{\frac{1}{2} \cdot re}{{im}^{2}}}\right) + \frac{re}{{im}^{4}}\right) \]

       12. *-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \frac{\color{blue}{re \cdot \frac{1}{2}}}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       13. associate-/l* N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \color{blue}{re \cdot \frac{\frac{1}{2}}{{im}^{2}}}\right) + \frac{re}{{im}^{4}}\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + re \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       15. associate-*r/ N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right) + \frac{re}{{im}^{4}}\right) \]

       16. distribute-lft-out N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{re \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)} + \frac{re}{{im}^{4}}\right) \]

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

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}, \frac{re}{{im}^{4}}\right)} \]

   11. Simplified 49.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664 + \frac{0.5}{im \cdot im}, \frac{re}{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot re\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 49.9

          \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot re\right)} \]

   14. Simplified 49.9%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot re\right), -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot 0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (* re (* (fma 0.5 (* im im) 1.0) (fma -0.16666666666666666 (* re re) 1.0)))
   (* (* (* im im) (* im im)) (* re 0.041666666666666664))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = re * (fma(0.5, (im * im), 1.0) * fma(-0.16666666666666666, (re * re), 1.0));
	} else {
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = Float64(re * Float64(fma(0.5, Float64(im * im), 1.0) * fma(-0.16666666666666666, Float64(re * re), 1.0)));
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(re * 0.041666666666666664));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 80.2

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

   7. Simplified 80.2%

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

   8. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 61.2

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

   10. Simplified 61.2%

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

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 87.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

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

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot re \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right)} \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, 1\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot re \]

      11. *-lowering-*.f64 48.7

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right), 1\right) \cdot re \]

   8. Simplified 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot re} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right)} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       9. associate-+r+ N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot re + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right)} \]

       10. *-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(\color{blue}{re \cdot \frac{1}{24}} + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       11. associate-*r/ N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \color{blue}{\frac{\frac{1}{2} \cdot re}{{im}^{2}}}\right) + \frac{re}{{im}^{4}}\right) \]

       12. *-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \frac{\color{blue}{re \cdot \frac{1}{2}}}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       13. associate-/l* N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \color{blue}{re \cdot \frac{\frac{1}{2}}{{im}^{2}}}\right) + \frac{re}{{im}^{4}}\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + re \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       15. associate-*r/ N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right) + \frac{re}{{im}^{4}}\right) \]

       16. distribute-lft-out N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{re \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)} + \frac{re}{{im}^{4}}\right) \]

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

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}, \frac{re}{{im}^{4}}\right)} \]

   11. Simplified 49.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664 + \frac{0.5}{im \cdot im}, \frac{re}{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot re\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 49.9

          \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot re\right)} \]

   14. Simplified 49.9%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot 0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) -0.05)
   (*
    (* re re)
    (* re (fma (* im im) -0.08333333333333333 -0.16666666666666666)))
   (fma (* im (* im (fma (* im im) 0.041666666666666664 0.5))) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= -0.05) {
		tmp = (re * re) * (re * fma((im * im), -0.08333333333333333, -0.16666666666666666));
	} else {
		tmp = fma((im * (im * fma((im * im), 0.041666666666666664, 0.5))), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.05)
		tmp = Float64(Float64(re * re) * Float64(re * fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666)));
	else
		tmp = fma(Float64(im * Float64(im * fma(Float64(im * im), 0.041666666666666664, 0.5))), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 64.7

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

   7. Simplified 64.7%

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

   8. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 29.9

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

   10. Simplified 29.9%

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

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \cdot \frac{-1}{6}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) \cdot \frac{-1}{6}\right)} \]

       3. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

       8. unpow2 N/A

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

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

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

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

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

       11. +-commutative N/A

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

       12. distribute-rgt-in N/A

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

       13. metadata-eval N/A

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. metadata-eval N/A

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}} + \frac{-1}{6}\right)\right) \]

       17. metadata-eval N/A

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

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

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

       19. unpow2 N/A

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

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

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

       21. metadata-eval 11.0

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, -0.16666666666666666\right)\right) \]

   13. Simplified 11.0%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)} \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 92.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

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

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot re \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right)} \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, 1\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot re \]

      11. *-lowering-*.f64 67.9

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right), 1\right) \cdot re \]

   8. Simplified 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot re} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{24} + \frac{1}{2}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\left(\left(im \cdot im\right) \cdot \frac{1}{24}\right) \cdot \left(im \cdot im\right) + \frac{1}{2} \cdot \left(im \cdot im\right)\right)} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{\left(im \cdot \left(im \cdot \frac{1}{24}\right)\right)} \cdot \left(im \cdot im\right) + \frac{1}{2} \cdot \left(im \cdot im\right)\right) + 1\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right) + \frac{1}{2}\right)} + 1\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right) + \frac{1}{2}\right)\right) \cdot re + 1 \cdot re} \]

      6. *-lft-identity N/A

         \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right) + \frac{1}{2}\right)\right) \cdot re + \color{blue}{re} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right) + \frac{1}{2}\right), re, re\right)} \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right) + \frac{1}{2}\right)\right)}, re, re\right) \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right) + \frac{1}{2}\right)\right)}, re, re\right) \]

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

          \[\leadsto \mathsf{fma}\left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \frac{1}{24}\right) + \frac{1}{2}\right)\right)}, re, re\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(im \cdot \left(im \cdot \left(\color{blue}{\left(im \cdot im\right) \cdot \frac{1}{24}} + \frac{1}{2}\right)\right), re, re\right) \]

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

          \[\leadsto \mathsf{fma}\left(im \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right)}\right), re, re\right) \]

      13. *-lowering-*.f64 67.9

          \[\leadsto \mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right)\right), re, re\right) \]

   10. Applied egg-rr 67.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right), re, re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 45.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) -0.05)
   (*
    (* re re)
    (* re (fma (* im im) -0.08333333333333333 -0.16666666666666666)))
   (* re (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= -0.05) {
		tmp = (re * re) * (re * fma((im * im), -0.08333333333333333, -0.16666666666666666));
	} else {
		tmp = re * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.05)
		tmp = Float64(Float64(re * re) * Float64(re * fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666)));
	else
		tmp = Float64(re * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 64.7

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

   7. Simplified 64.7%

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

   8. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 29.9

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

   10. Simplified 29.9%

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

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \cdot \frac{-1}{6}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) \cdot \frac{-1}{6}\right)} \]

       3. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

       8. unpow2 N/A

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

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

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

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

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

       11. +-commutative N/A

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

       12. distribute-rgt-in N/A

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

       13. metadata-eval N/A

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. metadata-eval N/A

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}} + \frac{-1}{6}\right)\right) \]

       17. metadata-eval N/A

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

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

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

       19. unpow2 N/A

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

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

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

       21. metadata-eval 11.0

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, -0.16666666666666666\right)\right) \]

   13. Simplified 11.0%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)} \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 92.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

   8. Simplified 67.9%

      \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot 0.041666666666666664\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) -0.05)
   (*
    (* re re)
    (* re (fma (* im im) -0.08333333333333333 -0.16666666666666666)))
   (* re (fma (* im im) (* im (* im 0.041666666666666664)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= -0.05) {
		tmp = (re * re) * (re * fma((im * im), -0.08333333333333333, -0.16666666666666666));
	} else {
		tmp = re * fma((im * im), (im * (im * 0.041666666666666664)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.05)
		tmp = Float64(Float64(re * re) * Float64(re * fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666)));
	else
		tmp = Float64(re * fma(Float64(im * im), Float64(im * Float64(im * 0.041666666666666664)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

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

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \cdot \sin re \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \cdot \sin re \]

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

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

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      12. exp-0 N/A

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

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

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 64.7

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

   7. Simplified 64.7%

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

   8. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 29.9

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

   10. Simplified 29.9%

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

   11. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({re}^{3} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({re}^{3} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right) \cdot \frac{-1}{6}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\left(1 + \frac{1}{2} \cdot {im}^{2}\right) \cdot \frac{-1}{6}\right)} \]

       3. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)\right)} \]

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

       8. unpow2 N/A

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

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

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

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

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

       11. +-commutative N/A

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

       12. distribute-rgt-in N/A

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

       13. metadata-eval N/A

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

       14. *-commutative N/A

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

       15. associate-*l* N/A

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

       16. metadata-eval N/A

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \left({im}^{2} \cdot \color{blue}{\frac{-1}{12}} + \frac{-1}{6}\right)\right) \]

       17. metadata-eval N/A

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

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

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

       19. unpow2 N/A

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

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

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

       21. metadata-eval 11.0

           \[\leadsto \left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, -0.16666666666666666\right)\right) \]

   13. Simplified 11.0%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)} \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 92.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

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

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot re \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right)} \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, 1\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot re \]

      11. *-lowering-*.f64 67.9

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right), 1\right) \cdot re \]

   8. Simplified 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot re} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2}}, 1\right) \cdot re \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}}, 1\right) \cdot re \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24}, 1\right) \cdot re \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \frac{1}{24}\right)}, 1\right) \cdot re \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{1}{24} \cdot im\right)}, 1\right) \cdot re \]

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

          \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(\frac{1}{24} \cdot im\right)}, 1\right) \cdot re \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}, 1\right) \cdot re \]

       7. *-lowering-*.f64 67.8

          \[\leadsto \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(im \cdot 0.041666666666666664\right)}, 1\right) \cdot re \]

   11. Simplified 67.8%

       \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot 0.041666666666666664\right)}, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot 0.041666666666666664\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 44.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot -0.16666666666666666\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot 0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (fma (* re (* re -0.16666666666666666)) re re)
   (* (* (* im im) (* im im)) (* re 0.041666666666666664))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = fma((re * (re * -0.16666666666666666)), re, re);
	} else {
		tmp = ((im * im) * (im * im)) * (re * 0.041666666666666664);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = fma(Float64(re * Float64(re * -0.16666666666666666)), re, re);
	else
		tmp = Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(re * 0.041666666666666664));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 66.9

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   9. Taylor expanded in im around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 50.8

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

   11. Simplified 50.8%

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

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

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 50.8

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

   13. Applied egg-rr 50.8%

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

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 87.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

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

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot re \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right)} \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, 1\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot re \]

      11. *-lowering-*.f64 48.7

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right), 1\right) \cdot re \]

   8. Simplified 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot re} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right)} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       9. associate-+r+ N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot re + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right)} \]

       10. *-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(\color{blue}{re \cdot \frac{1}{24}} + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       11. associate-*r/ N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \color{blue}{\frac{\frac{1}{2} \cdot re}{{im}^{2}}}\right) + \frac{re}{{im}^{4}}\right) \]

       12. *-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \frac{\color{blue}{re \cdot \frac{1}{2}}}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       13. associate-/l* N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \color{blue}{re \cdot \frac{\frac{1}{2}}{{im}^{2}}}\right) + \frac{re}{{im}^{4}}\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + re \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       15. associate-*r/ N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right) + \frac{re}{{im}^{4}}\right) \]

       16. distribute-lft-out N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{re \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)} + \frac{re}{{im}^{4}}\right) \]

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

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}, \frac{re}{{im}^{4}}\right)} \]

   11. Simplified 49.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664 + \frac{0.5}{im \cdot im}, \frac{re}{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot re\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 49.9

          \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot re\right)} \]

   14. Simplified 49.9%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(0.041666666666666664 \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot -0.16666666666666666\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(re \cdot 0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 44.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot -0.16666666666666666\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (fma (* re (* re -0.16666666666666666)) re re)
   (* re (* im (* im (* im (* im 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = fma((re * (re * -0.16666666666666666)), re, re);
	} else {
		tmp = re * (im * (im * (im * (im * 0.041666666666666664))));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = fma(Float64(re * Float64(re * -0.16666666666666666)), re, re);
	else
		tmp = Float64(re * Float64(im * Float64(im * Float64(im * Float64(im * 0.041666666666666664)))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], N[(re * N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 66.9

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   9. Taylor expanded in im around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 50.8

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

   11. Simplified 50.8%

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

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

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 50.8

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

   13. Applied egg-rr 50.8%

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

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 87.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

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

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot re \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right)} \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, 1\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot re \]

      11. *-lowering-*.f64 48.7

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right), 1\right) \cdot re \]

   8. Simplified 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot re} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \cdot re \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot re \]

       2. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \cdot re \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \cdot re \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \cdot re \]

       5. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \cdot re \]

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

          \[\leadsto \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \cdot re \]

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

          \[\leadsto \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \cdot re \]

       9. *-commutative N/A

          \[\leadsto \left(im \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)}\right)\right) \cdot re \]

       10. unpow2 N/A

           \[\leadsto \left(im \cdot \left(im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24}\right)\right)\right) \cdot re \]

       11. associate-*l* N/A

           \[\leadsto \left(im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot \frac{1}{24}\right)\right)}\right)\right) \cdot re \]

       12. *-commutative N/A

           \[\leadsto \left(im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(\frac{1}{24} \cdot im\right)}\right)\right)\right) \cdot re \]

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

           \[\leadsto \left(im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot im\right)\right)}\right)\right) \cdot re \]

       14. *-commutative N/A

           \[\leadsto \left(im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(im \cdot \frac{1}{24}\right)}\right)\right)\right) \cdot re \]

       15. *-lowering-*.f64 49.0

           \[\leadsto \left(im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(im \cdot 0.041666666666666664\right)}\right)\right)\right) \cdot re \]

   11. Simplified 49.0%

       \[\leadsto \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)} \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot -0.16666666666666666\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 42.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot -0.16666666666666666\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (fma (* re (* re -0.16666666666666666)) re re)
   (* im (* im (* re (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = fma((re * (re * -0.16666666666666666)), re, re);
	} else {
		tmp = im * (im * (re * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = fma(Float64(re * Float64(re * -0.16666666666666666)), re, re);
	else
		tmp = Float64(im * Float64(im * Float64(re * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], N[(im * N[(im * N[(re * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 66.9

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   9. Taylor expanded in im around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 50.8

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

   11. Simplified 50.8%

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

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

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 50.8

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

   13. Applied egg-rr 50.8%

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

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 87.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

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

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot re \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right)} \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, 1\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot re \]

      11. *-lowering-*.f64 48.7

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right), 1\right) \cdot re \]

   8. Simplified 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot re} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right)} \]
   10. Step-by-step derivation

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

          \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto {im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

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

          \[\leadsto \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \left(\frac{1}{24} \cdot re + \left(\frac{1}{2} \cdot \frac{re}{{im}^{2}} + \frac{re}{{im}^{4}}\right)\right) \]

       9. associate-+r+ N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot re + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right)} \]

       10. *-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(\color{blue}{re \cdot \frac{1}{24}} + \frac{1}{2} \cdot \frac{re}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       11. associate-*r/ N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \color{blue}{\frac{\frac{1}{2} \cdot re}{{im}^{2}}}\right) + \frac{re}{{im}^{4}}\right) \]

       12. *-commutative N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \frac{\color{blue}{re \cdot \frac{1}{2}}}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       13. associate-/l* N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + \color{blue}{re \cdot \frac{\frac{1}{2}}{{im}^{2}}}\right) + \frac{re}{{im}^{4}}\right) \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + re \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{{im}^{2}}\right) + \frac{re}{{im}^{4}}\right) \]

       15. associate-*r/ N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(re \cdot \frac{1}{24} + re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right) + \frac{re}{{im}^{4}}\right) \]

       16. distribute-lft-out N/A

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\color{blue}{re \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)} + \frac{re}{{im}^{4}}\right) \]

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

           \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}, \frac{re}{{im}^{4}}\right)} \]

   11. Simplified 49.0%

       \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \mathsf{fma}\left(re, 0.041666666666666664 + \frac{0.5}{im \cdot im}, \frac{re}{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot re} \]

       2. metadata-eval N/A

          \[\leadsto \left(\frac{1}{24} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot re \]

       3. pow-sqr N/A

          \[\leadsto \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \cdot re \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \cdot re \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \cdot re \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re\right)} \]

       7. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)} \]

       8. unpow2 N/A

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)} \]

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

           \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)} \]

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

           \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right)\right)\right)} \]

       12. associate-*r* N/A

           \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re\right)}\right) \]

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

           \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot re\right)}\right) \]

       14. *-commutative N/A

           \[\leadsto im \cdot \left(im \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)} \cdot re\right)\right) \]

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

           \[\leadsto im \cdot \left(im \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)} \cdot re\right)\right) \]

       16. unpow2 N/A

           \[\leadsto im \cdot \left(im \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24}\right) \cdot re\right)\right) \]

       17. *-lowering-*.f64 47.2

           \[\leadsto im \cdot \left(im \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot 0.041666666666666664\right) \cdot re\right)\right) \]

   14. Simplified 47.2%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\left(\left(im \cdot im\right) \cdot 0.041666666666666664\right) \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot -0.16666666666666666\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 40.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot -0.16666666666666666\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (fma (* re (* re -0.16666666666666666)) re re)
   (* re (fma (* im im) 0.5 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = fma((re * (re * -0.16666666666666666)), re, re);
	} else {
		tmp = re * fma((im * im), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = fma(Float64(re * Float64(re * -0.16666666666666666)), re, re);
	else
		tmp = Float64(re * fma(Float64(im * im), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 66.9

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   9. Taylor expanded in im around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 50.8

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

   11. Simplified 50.8%

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

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

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 50.8

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

   13. Applied egg-rr 50.8%

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

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 87.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

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

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot re \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right)} \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, 1\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot re \]

      11. *-lowering-*.f64 48.7

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right), 1\right) \cdot re \]

   8. Simplified 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot re} \]

   9. Taylor expanded in im around 0

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

   11. Simplified 43.2%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot -0.16666666666666666\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 40.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;re \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) 0.0001)
   (* re (fma -0.16666666666666666 (* re re) 1.0))
   (* re (fma (* im im) 0.5 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= 0.0001) {
		tmp = re * fma(-0.16666666666666666, (re * re), 1.0);
	} else {
		tmp = re * fma((im * im), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0001)
		tmp = Float64(re * fma(-0.16666666666666666, Float64(re * re), 1.0));
	else
		tmp = Float64(re * fma(Float64(im * im), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 66.9

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 66.9%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   9. Taylor expanded in im around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 50.8

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

   11. Simplified 50.8%

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

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 87.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

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

         \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot re \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right)} \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, 1\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, 1\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot re \]

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

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot re \]

      11. *-lowering-*.f64 48.7

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right), 1\right) \cdot re \]

   8. Simplified 48.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot re} \]

   9. Taylor expanded in im around 0

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

   11. Simplified 43.2%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;re \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 29.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- 0.0 im)) (exp im))) -0.05)
   (* -0.16666666666666666 (* re (* re re)))
   re))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= -0.05) {
		tmp = -0.16666666666666666 * (re * (re * re));
	} else {
		tmp = re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((sin(re) * 0.5d0) * (exp((0.0d0 - im)) + exp(im))) <= (-0.05d0)) then
        tmp = (-0.16666666666666666d0) * (re * (re * re))
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((Math.sin(re) * 0.5) * (Math.exp((0.0 - im)) + Math.exp(im))) <= -0.05) {
		tmp = -0.16666666666666666 * (re * (re * re));
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((math.sin(re) * 0.5) * (math.exp((0.0 - im)) + math.exp(im))) <= -0.05:
		tmp = -0.16666666666666666 * (re * (re * re))
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.05)
		tmp = Float64(-0.16666666666666666 * Float64(re * Float64(re * re)));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((sin(re) * 0.5) * (exp((0.0 - im)) + exp(im))) <= -0.05)
		tmp = -0.16666666666666666 * (re * (re * re));
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 85.8%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-lowering-*.f64 40.7

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 40.7%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   9. Taylor expanded in im around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 10.5

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

   11. Simplified 10.5%

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

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   13. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 10.2

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

   14. Simplified 10.2%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 62.1

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

   5. Simplified 62.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \]
   7. Step-by-step derivation

   8. Simplified 40.9%

      \[\leadsto \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-6}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 1e-6)
   (*
    re
    (*
     (fma re (* re -0.16666666666666666) 1.0)
     (fma
      (* im im)
      (fma
       (* im im)
       (fma (* im im) 0.001388888888888889 0.041666666666666664)
       0.5)
      1.0)))
   (*
    (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0)
    (*
     re
     (fma
      (* re re)
      (fma (* re re) 0.008333333333333333 -0.16666666666666666)
      1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 1e-6) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma((im * im), fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0));
	} else {
		tmp = fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0) * (re * fma((re * re), fma((re * re), 0.008333333333333333, -0.16666666666666666), 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 1e-6)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0)));
	else
		tmp = Float64(fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0) * Float64(re * fma(Float64(re * re), fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 1e-6], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 9.99999999999999955e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \sin re} \]

      2. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} + \sin re \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 91.4%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 70.5%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)} \]

   if 9.99999999999999955e-7 < (sin.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      11. unpow2 N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      12. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      13. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      10. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      11. *-lowering-*.f64 38.4

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

   8. Simplified 38.4%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-6}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 33.7% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ re \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* re (fma -0.16666666666666666 (* re re) 1.0)))

C

double code(double re, double im) {
	return re * fma(-0.16666666666666666, (re * re), 1.0);
}

Julia

function code(re, im)
	return Float64(re * fma(-0.16666666666666666, Float64(re * re), 1.0))
end

Wolfram

code[re_, im_] := N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

   4. associate-*r* N/A

      \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

   6. *-commutative N/A

      \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} + \sin re\right) \]

   7. associate-*r* N/A

      \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2}} + \sin re\right) \]

   8. *-commutative N/A

      \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

   9. associate-*r* N/A

      \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   11. unpow2 N/A

       \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

   12. associate-*r* N/A

       \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

   13. *-commutative N/A

       \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

5. Simplified 90.0%

   \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]
7. Step-by-step derivation

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

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

   2. +-commutative N/A

      \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

   3. *-commutative N/A

      \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

   4. unpow2 N/A

      \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

   5. associate-*l* N/A

      \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

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

      \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

   7. *-lowering-*.f64 53.0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot -0.16666666666666666}, 1\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

8. Simplified 53.0%

   \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]

9. Taylor expanded in im around 0

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

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

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

    2. +-commutative N/A

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

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

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

    4. unpow2 N/A

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

    5. *-lowering-*.f64 33.4

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

11. Simplified 33.4%

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

Alternative 29: 26.3% accurate, 317.0× speedup

Math

\[\begin{array}{l} \\ re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 re)

C

double code(double re, double im) {
	return re;
}

Fortran

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

Java

public static double code(double re, double im) {
	return re;
}

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

function tmp = code(re, im)
	tmp = re;
end

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. sin-lowering-sin.f64 52.3

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

5. Simplified 52.3%

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

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re} \]
7. Step-by-step derivation

8. Simplified 28.8%

   \[\leadsto \color{blue}{re} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))