math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 16

Speedup: 1.5×

• 48.7% 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, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \sin re}{\frac{0.5}{\cosh im}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (* 0.5 (sin re)) (/ 0.5 (cosh im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) / (0.5 / cosh(im));
}

Fortran

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

Java

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

Python

def code(re, im):
	return (0.5 * math.sin(re)) / (0.5 / math.cosh(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) / Float64(0.5 / cosh(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) / (0.5 / cosh(im));
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip3-+ N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}} \]

   4. clear-num N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{1}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

   10. clear-num N/A

       \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\color{blue}{\frac{1}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}}}} \]

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\sin re \cdot 0.5}{\frac{1}{\cosh im \cdot 2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 99.6

      \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{0.5}{\cosh im}}} \]

6. Applied rewrites 99.6%

   \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{0.5}{\cosh im}}} \]

7. Final simplification 99.6%

   \[\leadsto \frac{0.5 \cdot \sin re}{\frac{0.5}{\cosh im}} \]
8. Add Preprocessing

Alternative 2: 79.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := \left(e^{im} + e^{-im}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{re}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* (+ (exp im) (exp (- im))) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma im im 2.0)
      (*
       (fma
        (fma
         (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
         (* re re)
         -0.08333333333333333)
        (* re re)
        0.5)
       re))
     (if (<= t_1 1.0)
       (* (fma im im 2.0) t_0)
       (*
        (/ 1.0 (/ 1.0 re))
        (fma
         (fma
          (fma 0.001388888888888889 (* im im) 0.041666666666666664)
          (* im im)
          0.5)
         (* im im)
         1.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = (exp(im) + exp(-im)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(im, im, 2.0) * (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re);
	} else if (t_1 <= 1.0) {
		tmp = fma(im, im, 2.0) * t_0;
	} else {
		tmp = (1.0 / (1.0 / re)) * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(Float64(exp(im) + exp(Float64(-im))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(im, im, 2.0) * Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re));
	elseif (t_1 <= 1.0)
		tmp = Float64(fma(im, im, 2.0) * t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 / re)) * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(im * im + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(1.0 / N[(1.0 / re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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. lower-fma.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 54.3

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 54.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 54.3%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) + \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       6. sub-neg N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       8. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{12}}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       9. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}, {re}^{2}, \frac{-1}{12}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       10. +-commutative N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{10080} \cdot {re}^{2} + \frac{1}{240}}, {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{10080}, {re}^{2}, \frac{1}{240}\right)}, {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       12. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, \color{blue}{re \cdot re}, \frac{1}{240}\right), {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, \color{blue}{re \cdot re}, \frac{1}{240}\right), {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       14. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), \color{blue}{re \cdot re}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), \color{blue}{re \cdot re}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       16. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), re \cdot re, \frac{-1}{12}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       17. lower-*.f64 58.4

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), \color{blue}{re \cdot re}, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   13. Applied rewrites 58.4%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   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. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   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 98.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.0

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

   7. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 91.0

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

   9. Applied rewrites 91.0%

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

   10. Taylor expanded in re around 0

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

       1. lower-/.f64 74.0

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

   12. Applied rewrites 74.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{re}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;1 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{re}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp im) (exp (- im))) (* 0.5 (sin re)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma im im 2.0)
      (*
       (fma
        (fma
         (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
         (* re re)
         -0.08333333333333333)
        (* re re)
        0.5)
       re))
     (if (<= t_0 1.0)
       (* 1.0 (sin re))
       (*
        (/ 1.0 (/ 1.0 re))
        (fma
         (fma
          (fma 0.001388888888888889 (* im im) 0.041666666666666664)
          (* im im)
          0.5)
         (* im im)
         1.0))))))

C

double code(double re, double im) {
	double t_0 = (exp(im) + exp(-im)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(im, im, 2.0) * (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re);
	} else if (t_0 <= 1.0) {
		tmp = 1.0 * sin(re);
	} else {
		tmp = (1.0 / (1.0 / re)) * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(im, im, 2.0) * Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re));
	elseif (t_0 <= 1.0)
		tmp = Float64(1.0 * sin(re));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 / re)) * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(1.0 * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 / re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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. lower-fma.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 54.3

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 54.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 54.3%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) + \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       6. sub-neg N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       8. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{12}}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       9. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}, {re}^{2}, \frac{-1}{12}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       10. +-commutative N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{10080} \cdot {re}^{2} + \frac{1}{240}}, {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{10080}, {re}^{2}, \frac{1}{240}\right)}, {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       12. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, \color{blue}{re \cdot re}, \frac{1}{240}\right), {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, \color{blue}{re \cdot re}, \frac{1}{240}\right), {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       14. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), \color{blue}{re \cdot re}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), \color{blue}{re \cdot re}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       16. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), re \cdot re, \frac{-1}{12}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       17. lower-*.f64 58.4

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), \color{blue}{re \cdot re}, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   13. Applied rewrites 58.4%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1} \cdot \sin re \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto \color{blue}{1} \cdot \sin re \]

   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 98.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.0

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

   7. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 91.0

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

   9. Applied rewrites 91.0%

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

   10. Taylor expanded in re around 0

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

       1. lower-/.f64 74.0

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

   12. Applied rewrites 74.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 1:\\ \;\;\;\;1 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{re}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.7

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

   7. Applied rewrites 93.7%

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

   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 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}} \]

      4. clear-num N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{1}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

      10. clear-num N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\color{blue}{\frac{1}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}}}} \]

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{\sin re \cdot 0.5}{\frac{1}{\cosh im \cdot 2}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 98.2

         \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{0.5}{\cosh im}}} \]

   6. Applied rewrites 98.2%

      \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{0.5}{\cosh im}}} \]

   7. Taylor expanded in re around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot re}}{\frac{\frac{1}{2}}{\cosh im}} \]
   8. Step-by-step derivation

      1. lower-*.f64 79.3

         \[\leadsto \frac{\color{blue}{0.5 \cdot re}}{\frac{0.5}{\cosh im}} \]

   9. Applied rewrites 79.3%

      \[\leadsto \frac{\color{blue}{0.5 \cdot re}}{\frac{0.5}{\cosh im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

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

Alternative 5: 89.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.7

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {im}^{2}, im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot \sin re \]
   9. Step-by-step derivation

   10. Applied rewrites 93.4%

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

   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 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}} \]

      4. clear-num N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{1}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

      10. clear-num N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\color{blue}{\frac{1}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}}}} \]

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{\sin re \cdot 0.5}{\frac{1}{\cosh im \cdot 2}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 98.2

         \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{0.5}{\cosh im}}} \]

   6. Applied rewrites 98.2%

      \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{0.5}{\cosh im}}} \]

   7. Taylor expanded in re around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot re}}{\frac{\frac{1}{2}}{\cosh im}} \]
   8. Step-by-step derivation

      1. lower-*.f64 79.3

         \[\leadsto \frac{\color{blue}{0.5 \cdot re}}{\frac{0.5}{\cosh im}} \]

   9. Applied rewrites 79.3%

      \[\leadsto \frac{\color{blue}{0.5 \cdot re}}{\frac{0.5}{\cosh im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

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

Alternative 6: 88.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 90.2

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

   7. Applied rewrites 90.2%

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

   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 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}} \]

      4. clear-num N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{1}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

      10. clear-num N/A

          \[\leadsto \frac{\sin re \cdot \frac{1}{2}}{\color{blue}{\frac{1}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}}}} \]

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{\sin re \cdot 0.5}{\frac{1}{\cosh im \cdot 2}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 98.2

         \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{0.5}{\cosh im}}} \]

   6. Applied rewrites 98.2%

      \[\leadsto \frac{\sin re \cdot 0.5}{\color{blue}{\frac{0.5}{\cosh im}}} \]

   7. Taylor expanded in re around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot re}}{\frac{\frac{1}{2}}{\cosh im}} \]
   8. Step-by-step derivation

      1. lower-*.f64 79.3

         \[\leadsto \frac{\color{blue}{0.5 \cdot re}}{\frac{0.5}{\cosh im}} \]

   9. Applied rewrites 79.3%

      \[\leadsto \frac{\color{blue}{0.5 \cdot re}}{\frac{0.5}{\cosh im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

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

Alternative 7: 85.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 90.2

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

   7. Applied rewrites 90.2%

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

   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 98.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 91.0

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

   7. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 91.0

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

   9. Applied rewrites 91.0%

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

   10. Taylor expanded in re around 0

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

       1. lower-/.f64 74.0

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

   12. Applied rewrites 74.0%

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

4. Final simplification 86.9%

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

Alternative 8: 54.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{re}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) 0.004)
   (*
    (fma im im 2.0)
    (*
     (fma
      (fma
       (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
       (* re re)
       -0.08333333333333333)
      (* re re)
      0.5)
     re))
   (*
    (/ 1.0 (/ 1.0 re))
    (fma
     (fma
      (fma 0.001388888888888889 (* im im) 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= 0.004) {
		tmp = fma(im, im, 2.0) * (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re);
	} else {
		tmp = (1.0 / (1.0 / re)) * fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= 0.004)
		tmp = Float64(fma(im, im, 2.0) * Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 / re)) * fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 / re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $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))) < 0.0040000000000000001

   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. lower-fma.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 62.8

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 62.8%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.8%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) + \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       6. sub-neg N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       8. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{12}}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       9. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}, {re}^{2}, \frac{-1}{12}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       10. +-commutative N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{10080} \cdot {re}^{2} + \frac{1}{240}}, {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{10080}, {re}^{2}, \frac{1}{240}\right)}, {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       12. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, \color{blue}{re \cdot re}, \frac{1}{240}\right), {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, \color{blue}{re \cdot re}, \frac{1}{240}\right), {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       14. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), \color{blue}{re \cdot re}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), \color{blue}{re \cdot re}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       16. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), re \cdot re, \frac{-1}{12}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       17. lower-*.f64 64.5

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), \color{blue}{re \cdot re}, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   13. Applied rewrites 64.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 0.0040000000000000001 < (*.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 98.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 94.3

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

   7. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 94.2

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

   9. Applied rewrites 94.2%

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

   10. Taylor expanded in re around 0

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

       1. lower-/.f64 48.1

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

   12. Applied rewrites 48.1%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{re}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.004:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) 0.004)
   (* (* (fma -0.08333333333333333 (* re re) 0.5) re) (fma im im 2.0))
   (* (* 0.5 re) (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= 0.004) {
		tmp = (fma(-0.08333333333333333, (re * re), 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= 0.004)
		tmp = Float64(Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $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))) < 0.0040000000000000001

   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. lower-fma.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 62.8

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 62.8%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.8%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 62.8

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

   13. Applied rewrites 62.8%

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

   if 0.0040000000000000001 < (*.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 98.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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. lower-fma.f64 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 25.3

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 25.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 25.3%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

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

       1. lower-*.f64 27.4

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

   13. Applied rewrites 27.4%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.004:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.004:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right) \cdot 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) 0.004)
   (* (* (fma (* -0.16666666666666666 (* re re)) re re) 0.5) 2.0)
   (* (* 0.5 re) (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= 0.004) {
		tmp = (fma((-0.16666666666666666 * (re * re)), re, re) * 0.5) * 2.0;
	} else {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= 0.004)
		tmp = Float64(Float64(fma(Float64(-0.16666666666666666 * Float64(re * re)), re, re) * 0.5) * 2.0);
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $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))) < 0.0040000000000000001

   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. lower-fma.f64 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 62.8

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 62.8%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.8%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in im around 0

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

   13. Applied rewrites 47.9%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right)\right) \cdot \color{blue}{2} \]

   if 0.0040000000000000001 < (*.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 98.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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. lower-fma.f64 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 25.3

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 25.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 25.3%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

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

       1. lower-*.f64 27.4

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

   13. Applied rewrites 27.4%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.004:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right) \cdot 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub0-neg N/A

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

   13. cosh-undef N/A

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

   14. associate-*r* N/A

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

   15. metadata-eval N/A

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

   16. exp-0 N/A

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

   17. lower-*.f64 N/A

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

   18. exp-0 N/A

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

   19. lower-cosh.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 12: 50.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.04)
   (*
    (fma im im 2.0)
    (*
     (fma
      (fma
       (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
       (* re re)
       -0.08333333333333333)
      (* re re)
      0.5)
     re))
   (*
    (*
     (fma
      (fma 0.004166666666666667 (* re re) -0.08333333333333333)
      (* re re)
      0.5)
     re)
    (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.04) {
		tmp = fma(im, im, 2.0) * (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re);
	} else {
		tmp = (fma(fma(0.004166666666666667, (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.04)
		tmp = Float64(fma(im, im, 2.0) * Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re));
	else
		tmp = Float64(Float64(fma(fma(0.004166666666666667, Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.04], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0400000000000000008

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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. lower-fma.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 26.0

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 26.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 26.0%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) + \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       6. sub-neg N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       7. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       8. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{12}}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       9. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}, {re}^{2}, \frac{-1}{12}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       10. +-commutative N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{10080} \cdot {re}^{2} + \frac{1}{240}}, {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{10080}, {re}^{2}, \frac{1}{240}\right)}, {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       12. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, \color{blue}{re \cdot re}, \frac{1}{240}\right), {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, \color{blue}{re \cdot re}, \frac{1}{240}\right), {re}^{2}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       14. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), \color{blue}{re \cdot re}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), \color{blue}{re \cdot re}, \frac{-1}{12}\right), {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       16. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{10080}, re \cdot re, \frac{1}{240}\right), re \cdot re, \frac{-1}{12}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       17. lower-*.f64 30.3

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), \color{blue}{re \cdot re}, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   13. Applied rewrites 30.3%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -0.0400000000000000008 < (sin.f64 re)

   1. Initial program 99.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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. lower-fma.f64 78.4

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 58.3

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 58.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 58.3%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 60.2

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

   13. Applied rewrites 60.2%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.04:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.04)
   (* (* (fma -0.08333333333333333 (* re re) 0.5) re) (fma im im 2.0))
   (*
    (*
     (fma
      (fma 0.004166666666666667 (* re re) -0.08333333333333333)
      (* re re)
      0.5)
     re)
    (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.04) {
		tmp = (fma(-0.08333333333333333, (re * re), 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = (fma(fma(0.004166666666666667, (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.04)
		tmp = Float64(Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(fma(fma(0.004166666666666667, Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.04], N[(N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0400000000000000008

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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. lower-fma.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 26.0

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 26.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 26.0%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 26.0

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

   13. Applied rewrites 26.0%

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

   if -0.0400000000000000008 < (sin.f64 re)

   1. Initial program 99.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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. lower-fma.f64 78.4

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 58.3

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 58.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 58.3%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 60.2

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

   13. Applied rewrites 60.2%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 49.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot re, re \cdot re, re\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 0.0002)
   (* (* (fma -0.08333333333333333 (* re re) 0.5) re) (fma im im 2.0))
   (*
    2.0
    (*
     (fma
      (* (fma (* re re) 0.008333333333333333 -0.16666666666666666) re)
      (* re re)
      re)
     0.5))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 0.0002) {
		tmp = (fma(-0.08333333333333333, (re * re), 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = 2.0 * (fma((fma((re * re), 0.008333333333333333, -0.16666666666666666) * re), (re * re), re) * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 0.0002)
		tmp = Float64(Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(2.0 * Float64(fma(Float64(fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666) * re), Float64(re * re), re) * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 0.0002], N[(N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * re), $MachinePrecision] * N[(re * re), $MachinePrecision] + re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 2.0000000000000001e-4

   1. Initial program 99.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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. lower-fma.f64 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 59.5

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 59.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.5%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 59.5

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

   13. Applied rewrites 59.5%

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

   if 2.0000000000000001e-4 < (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

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. cube-unmult N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re\right)\right) \cdot 2 \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re\right)\right) \cdot 2 \]

      9. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left({re}^{3}, \color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, re\right)\right) \cdot 2 \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left({re}^{3}, \frac{1}{120} \cdot {re}^{2} + \color{blue}{\frac{-1}{6}}, re\right)\right) \cdot 2 \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left({re}^{3}, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {re}^{2}, \frac{-1}{6}\right)}, re\right)\right) \cdot 2 \]

      12. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left({re}^{3}, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{re \cdot re}, \frac{-1}{6}\right), re\right)\right) \cdot 2 \]

      13. lower-*.f64 24.0

          \[\leadsto \left(0.5 \cdot \mathsf{fma}\left({re}^{3}, \mathsf{fma}\left(0.008333333333333333, \color{blue}{re \cdot re}, -0.16666666666666666\right), re\right)\right) \cdot 2 \]

   8. Applied rewrites 24.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right)}\right) \cdot 2 \]
   9. Step-by-step derivation

   10. Applied rewrites 24.0%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot re, \color{blue}{re \cdot re}, re\right)\right) \cdot 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot re, re \cdot re, re\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 0.004:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.008333333333333333 \cdot \left(re \cdot re\right)\right) \cdot re, re \cdot re, re\right) \cdot 0.5\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 0.004)
   (* (* (fma -0.08333333333333333 (* re re) 0.5) re) (fma im im 2.0))
   (*
    (* (fma (* (* 0.008333333333333333 (* re re)) re) (* re re) re) 0.5)
    2.0)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 0.004) {
		tmp = (fma(-0.08333333333333333, (re * re), 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = (fma(((0.008333333333333333 * (re * re)) * re), (re * re), re) * 0.5) * 2.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 0.004)
		tmp = Float64(Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re * re)) * re), Float64(re * re), re) * 0.5) * 2.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 0.004], N[(N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * N[(re * re), $MachinePrecision] + re), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 0.0040000000000000001

   1. Initial program 99.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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. lower-fma.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 59.6

         \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 59.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.6%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 59.6

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

   13. Applied rewrites 59.6%

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

   if 0.0040000000000000001 < (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

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. cube-unmult N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re\right)\right) \cdot 2 \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re\right)\right) \cdot 2 \]

      9. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left({re}^{3}, \color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, re\right)\right) \cdot 2 \]

      10. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left({re}^{3}, \frac{1}{120} \cdot {re}^{2} + \color{blue}{\frac{-1}{6}}, re\right)\right) \cdot 2 \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left({re}^{3}, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {re}^{2}, \frac{-1}{6}\right)}, re\right)\right) \cdot 2 \]

      12. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left({re}^{3}, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{re \cdot re}, \frac{-1}{6}\right), re\right)\right) \cdot 2 \]

      13. lower-*.f64 22.7

          \[\leadsto \left(0.5 \cdot \mathsf{fma}\left({re}^{3}, \mathsf{fma}\left(0.008333333333333333, \color{blue}{re \cdot re}, -0.16666666666666666\right), re\right)\right) \cdot 2 \]

   8. Applied rewrites 22.7%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right)}\right) \cdot 2 \]
   9. Step-by-step derivation

   10. Applied rewrites 22.7%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot re, \color{blue}{re \cdot re}, re\right)\right) \cdot 2 \]

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 22.7%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot 0.008333333333333333\right) \cdot re, re \cdot re, re\right)\right) \cdot 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 0.004:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.008333333333333333 \cdot \left(re \cdot re\right)\right) \cdot re, re \cdot re, re\right) \cdot 0.5\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.8% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 re) (fma im im 2.0)))

C

double code(double re, double im) {
	return (0.5 * re) * fma(im, im, 2.0);
}

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * fma(im, im, 2.0))
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
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. lower-fma.f64 77.5

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

5. Applied rewrites 77.5%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. cube-unmult N/A

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

   9. lower-pow.f64 50.5

      \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{{re}^{3}}, -0.16666666666666666, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

8. Applied rewrites 50.5%

   \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left({re}^{3}, -0.16666666666666666, re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
9. Step-by-step derivation

10. Applied rewrites 50.5%

    \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), \color{blue}{re}, re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

11. Taylor expanded in re around 0

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

    1. lower-*.f64 48.4

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

13. Applied rewrites 48.4%

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

Reproduce

herbie shell --seed 2024284 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))