math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 14

Speedup: 1.5×

• 49.5% 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, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \mathsf{fma}\left(0.5 \cdot \sin re, e^{-im\_m}, \left(e^{im\_m} \cdot 0.5\right) \cdot \sin re\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (fma (* 0.5 (sin re)) (exp (- im_m)) (* (* (exp im_m) 0.5) (sin re))))

C

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

Julia

im_m = abs(im)
function code(re, im_m)
	return fma(Float64(0.5 * sin(re)), exp(Float64(-im_m)), Float64(Float64(exp(im_m) * 0.5) * sin(re)))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[Exp[(-im$95$m)], $MachinePrecision] + N[(N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. 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. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub0-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(0.5 \cdot \sin re, e^{-im}, \left(e^{im} \cdot 0.5\right) \cdot \sin re\right) \]
6. Add Preprocessing

Alternative 2: 77.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := \left(e^{-im\_m} + e^{im\_m}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5} \cdot \left(re \cdot re\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{elif}\;t\_1 \leq 3:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m, 2\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({im\_m}^{4}, 0.08333333333333333, \mathsf{fma}\left(im\_m, im\_m, 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* (+ (exp (- im_m)) (exp im_m)) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (fma (* -9.92063492063492e-5 (* re re)) (* re re) -0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma im_m im_m 2.0))
     (if (<= t_1 3.0)
       (* (fma im_m im_m 2.0) t_0)
       (*
        (fma (pow im_m 4.0) 0.08333333333333333 (fma im_m im_m 2.0))
        (* 0.5 re))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double t_1 = (exp(-im_m) + exp(im_m)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma((-9.92063492063492e-5 * (re * re)), (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	} else if (t_1 <= 3.0) {
		tmp = fma(im_m, im_m, 2.0) * t_0;
	} else {
		tmp = fma(pow(im_m, 4.0), 0.08333333333333333, fma(im_m, im_m, 2.0)) * (0.5 * re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(Float64(exp(Float64(-im_m)) + exp(im_m)) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(-9.92063492063492e-5 * Float64(re * re)), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	elseif (t_1 <= 3.0)
		tmp = Float64(fma(im_m, im_m, 2.0) * t_0);
	else
		tmp = Float64(fma((im_m ^ 4.0), 0.08333333333333333, fma(im_m, im_m, 2.0)) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3.0], N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Power[im$95$m, 4.0], $MachinePrecision] * 0.08333333333333333 + N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $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 47.6

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

   5. Applied rewrites 47.6%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 52.7

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

   8. Applied rewrites 52.7%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. 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 52.8

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

   11. Applied rewrites 52.8%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 52.8%

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

   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))) < 3

   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 98.7

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

   5. Applied rewrites 98.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 47.8

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

   5. Applied rewrites 47.8%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 43.2

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

   8. Applied rewrites 43.2%

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

   9. Taylor expanded in re around 0

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

       1. lower-*.f64 48.4

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

   11. Applied rewrites 48.4%

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

   12. Taylor expanded in im around 0

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

       1. distribute-lft-in N/A

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

       2. *-rgt-identity N/A

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

       3. associate-+r+ N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. pow-sqr N/A

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

       8. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\frac{1}{12} \cdot {im}^{\color{blue}{4}} + \left(2 + {im}^{2}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\color{blue}{{im}^{4} \cdot \frac{1}{12}} + \left(2 + {im}^{2}\right)\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, \frac{1}{12}, 2 + {im}^{2}\right)} \]

       11. lower-pow.f64 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{4}}, \frac{1}{12}, 2 + {im}^{2}\right) \]

       12. +-commutative N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \frac{1}{12}, \color{blue}{{im}^{2} + 2}\right) \]

       13. unpow2 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \frac{1}{12}, \color{blue}{im \cdot im} + 2\right) \]

       14. lower-fma.f64 62.1

           \[\leadsto \left(0.5 \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, 0.08333333333333333, \color{blue}{\mathsf{fma}\left(im, im, 2\right)}\right) \]

   14. Applied rewrites 62.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5} \cdot \left(re \cdot re\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 3:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({im}^{4}, 0.08333333333333333, \mathsf{fma}\left(im, im, 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.4× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* (+ (exp (- im_m)) (exp im_m)) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (fma (* -9.92063492063492e-5 (* re re)) (* re re) -0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma im_m im_m 2.0))
     (if (<= t_1 5.0)
       (* (fma im_m im_m 2.0) t_0)
       (*
        (*
         (fma
          (fma 0.004166666666666667 (* re re) -0.08333333333333333)
          (* re re)
          0.5)
         re)
        (fma im_m im_m 2.0))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double t_1 = (exp(-im_m) + exp(im_m)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma((-9.92063492063492e-5 * (re * re)), (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	} else if (t_1 <= 5.0) {
		tmp = fma(im_m, im_m, 2.0) * t_0;
	} else {
		tmp = (fma(fma(0.004166666666666667, (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(Float64(exp(Float64(-im_m)) + exp(im_m)) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(-9.92063492063492e-5 * Float64(re * re)), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	elseif (t_1 <= 5.0)
		tmp = Float64(fma(im_m, im_m, 2.0) * t_0);
	else
		tmp = Float64(Float64(fma(fma(0.004166666666666667, Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5.0], N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * t$95$0), $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$95$m * im$95$m + 2.0), $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 47.6

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

   5. Applied rewrites 47.6%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 52.7

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

   8. Applied rewrites 52.7%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. 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 52.8

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

   11. Applied rewrites 52.8%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 52.8%

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

   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))) < 5

   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 47.8

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

   5. Applied rewrites 47.8%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 43.2

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

   8. Applied rewrites 43.2%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       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 54.2

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

   11. Applied rewrites 54.2%

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

4. Final simplification 75.7%

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

Alternative 4: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{-im\_m} + e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5} \cdot \left(re \cdot re\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{elif}\;t\_0 \leq 3:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (fma (* -9.92063492063492e-5 (* re re)) (* re re) -0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma im_m im_m 2.0))
     (if (<= t_0 3.0)
       (sin re)
       (*
        (*
         (fma
          (fma 0.004166666666666667 (* re re) -0.08333333333333333)
          (* re re)
          0.5)
         re)
        (fma im_m im_m 2.0))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(-im_m) + exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma((-9.92063492063492e-5 * (re * re)), (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	} else if (t_0 <= 3.0) {
		tmp = sin(re);
	} else {
		tmp = (fma(fma(0.004166666666666667, (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) + exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(-9.92063492063492e-5 * Float64(re * re)), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	elseif (t_0 <= 3.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(fma(fma(0.004166666666666667, Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 3.0], N[Sin[re], $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$95$m * im$95$m + 2.0), $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 47.6

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

   5. Applied rewrites 47.6%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 52.7

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

   8. Applied rewrites 52.7%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. 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 52.8

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

   11. Applied rewrites 52.8%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 52.8%

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

   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))) < 3

   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. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub0-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. distribute-lft-in N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. mul0-rgt N/A

         \[\leadsto \sin re + \color{blue}{0} \]

      7. +-rgt-identity N/A

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

      8. lower-sin.f64 98.6

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

   7. Applied rewrites 98.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 47.8

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

   5. Applied rewrites 47.8%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 43.2

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

   8. Applied rewrites 43.2%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       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 54.2

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

   11. Applied rewrites 54.2%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5} \cdot \left(re \cdot re\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 3:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} + e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 5 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666 \cdot re, re\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) 5e-6)
   (* 2.0 (* (fma (* re re) (* -0.16666666666666666 re) re) 0.5))
   (* (* 0.5 re) (fma im_m im_m 2.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(-im_m) + exp(im_m)) * (0.5 * sin(re))) <= 5e-6) {
		tmp = 2.0 * (fma((re * re), (-0.16666666666666666 * re), re) * 0.5);
	} else {
		tmp = (0.5 * re) * fma(im_m, im_m, 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) + exp(im_m)) * Float64(0.5 * sin(re))) <= 5e-6)
		tmp = Float64(2.0 * Float64(fma(Float64(re * re), Float64(-0.16666666666666666 * re), re) * 0.5));
	else
		tmp = Float64(Float64(0.5 * re) * fma(im_m, im_m, 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(2.0 * N[(N[(N[(re * re), $MachinePrecision] * N[(-0.16666666666666666 * re), $MachinePrecision] + re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * im$95$m + 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))) < 5.00000000000000041e-6

   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 76.9

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

   5. Applied rewrites 76.9%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 59.0

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

   8. Applied rewrites 59.0%

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

   10. Applied rewrites 59.0%

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

   11. Taylor expanded in im around 0

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

   13. Applied rewrites 43.3%

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

   if 5.00000000000000041e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 66.3

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

   5. Applied rewrites 66.3%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 28.8

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

   8. Applied rewrites 28.8%

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

   9. Taylor expanded in re around 0

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

       1. lower-*.f64 32.3

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

   11. Applied rewrites 32.3%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 5 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666 \cdot re, re\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \cosh im\_m \cdot \sin re \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* (cosh im_m) (sin re)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[Cosh[im$95$m], $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 100.0

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

   4. lift-*.f64 N/A

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

   5. *-lft-identity 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 7: 79.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 0.028:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right) \cdot \cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.001388888888888889, 0.041666666666666664\right), im\_m \cdot im\_m, 0.5\right) \cdot im\_m, im\_m, 1\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 0.028)
   (* (* 2.0 (* (fma -0.08333333333333333 (* re re) 0.5) re)) (cosh im_m))
   (*
    (fma
     (*
      (fma
       (fma (* im_m im_m) 0.001388888888888889 0.041666666666666664)
       (* im_m im_m)
       0.5)
      im_m)
     im_m
     1.0)
    (sin re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 0.028) {
		tmp = (2.0 * (fma(-0.08333333333333333, (re * re), 0.5) * re)) * cosh(im_m);
	} else {
		tmp = fma((fma(fma((im_m * im_m), 0.001388888888888889, 0.041666666666666664), (im_m * im_m), 0.5) * im_m), im_m, 1.0) * sin(re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 0.028)
		tmp = Float64(Float64(2.0 * Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re)) * cosh(im_m));
	else
		tmp = Float64(fma(Float64(fma(fma(Float64(im_m * im_m), 0.001388888888888889, 0.041666666666666664), Float64(im_m * im_m), 0.5) * im_m), im_m, 1.0) * sin(re));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 0.028], N[(N[(2.0 * N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.0280000000000000006

   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 re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 75.5

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)} \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right) \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right) \]

      7. neg-sub0 N/A

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

      8. cosh-undef N/A

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

      9. lift-cosh.f64 N/A

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

      10. *-lft-identity N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 75.5%

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

   if 0.0280000000000000006 < 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. 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 100.0

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 92.6

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

   9. Applied rewrites 92.6%

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

   11. Applied rewrites 92.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.028:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right) \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 0.028:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right) \cdot \cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im\_m \cdot im\_m, 0.041666666666666664\right), im\_m \cdot im\_m, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 0.028)
   (* (* 2.0 (* (fma -0.08333333333333333 (* re re) 0.5) re)) (cosh im_m))
   (*
    (fma
     (fma
      (fma 0.001388888888888889 (* im_m im_m) 0.041666666666666664)
      (* im_m im_m)
      0.5)
     (* im_m im_m)
     1.0)
    (sin re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 0.028) {
		tmp = (2.0 * (fma(-0.08333333333333333, (re * re), 0.5) * re)) * cosh(im_m);
	} else {
		tmp = fma(fma(fma(0.001388888888888889, (im_m * im_m), 0.041666666666666664), (im_m * im_m), 0.5), (im_m * im_m), 1.0) * sin(re);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 0.028)
		tmp = Float64(Float64(2.0 * Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re)) * cosh(im_m));
	else
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(im_m * im_m), 0.041666666666666664), Float64(im_m * im_m), 0.5), Float64(im_m * im_m), 1.0) * sin(re));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 0.028], N[(N[(2.0 * N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.001388888888888889 * N[(im$95$m * im$95$m), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.0280000000000000006

   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 re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 75.5

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)} \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right) \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right) \]

      7. neg-sub0 N/A

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

      8. cosh-undef N/A

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

      9. lift-cosh.f64 N/A

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

      10. *-lft-identity N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 75.5%

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

   if 0.0280000000000000006 < 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. 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 92.6

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

   7. Applied rewrites 92.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.028:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right) \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sin re) -0.01)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (fma im_m im_m 2.0))
   (*
    (*
     (fma
      (fma 0.004166666666666667 (* re re) -0.08333333333333333)
      (* re re)
      0.5)
     re)
    (fma im_m im_m 2.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(im_m, im_m, 2.0);
	} else {
		tmp = (fma(fma(0.004166666666666667, (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(Float64(fma(fma(0.004166666666666667, Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 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$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   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 72.8

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

   5. Applied rewrites 72.8%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 25.0

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

   8. Applied rewrites 25.0%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       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. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 25.0

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

   11. Applied rewrites 25.0%

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

   if -0.0100000000000000002 < (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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 54.0

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

   8. Applied rewrites 54.0%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       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 57.5

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

   11. Applied rewrites 57.5%

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

Alternative 10: 96.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right) \cdot im\_m, im\_m, 1\right) \cdot \sin re\\ \mathbf{if}\;im\_m \leq 1.75:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right) \cdot \cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0
         (*
          (fma (* (fma (* im_m im_m) 0.041666666666666664 0.5) im_m) im_m 1.0)
          (sin re))))
   (if (<= im_m 1.75)
     t_0
     (if (<= im_m 2.5e+77)
       (* (* 2.0 (* (fma -0.08333333333333333 (* re re) 0.5) re)) (cosh im_m))
       t_0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = fma((fma((im_m * im_m), 0.041666666666666664, 0.5) * im_m), im_m, 1.0) * sin(re);
	double tmp;
	if (im_m <= 1.75) {
		tmp = t_0;
	} else if (im_m <= 2.5e+77) {
		tmp = (2.0 * (fma(-0.08333333333333333, (re * re), 0.5) * re)) * cosh(im_m);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(fma(Float64(fma(Float64(im_m * im_m), 0.041666666666666664, 0.5) * im_m), im_m, 1.0) * sin(re))
	tmp = 0.0
	if (im_m <= 1.75)
		tmp = t_0;
	elseif (im_m <= 2.5e+77)
		tmp = Float64(Float64(2.0 * Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re)) * cosh(im_m));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 1.75], t$95$0, If[LessEqual[im$95$m, 2.5e+77], N[(N[(2.0 * N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.75 or 2.50000000000000002e77 < im

   1. Initial program 100.0%

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

      1. 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 92.5

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

   7. Applied rewrites 92.5%

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

   9. Applied rewrites 92.5%

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

   if 1.75 < im < 2.50000000000000002e77

   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 re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 73.7

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   5. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)} \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right) \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right) \]

      7. neg-sub0 N/A

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

      8. cosh-undef N/A

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

      9. lift-cosh.f64 N/A

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

      10. *-lft-identity N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 73.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right) \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot im, im, 1\right) \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m, im\_m, 2\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{if}\;im\_m \leq 1.75:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right) \cdot \cosh im\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (fma im_m im_m 2.0) (* 0.5 (sin re)))))
   (if (<= im_m 1.75)
     t_0
     (if (<= im_m 1.32e+154)
       (* (* 2.0 (* (fma -0.08333333333333333 (* re re) 0.5) re)) (cosh im_m))
       t_0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = fma(im_m, im_m, 2.0) * (0.5 * sin(re));
	double tmp;
	if (im_m <= 1.75) {
		tmp = t_0;
	} else if (im_m <= 1.32e+154) {
		tmp = (2.0 * (fma(-0.08333333333333333, (re * re), 0.5) * re)) * cosh(im_m);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(fma(im_m, im_m, 2.0) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (im_m <= 1.75)
		tmp = t_0;
	elseif (im_m <= 1.32e+154)
		tmp = Float64(Float64(2.0 * Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re)) * cosh(im_m));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 1.75], t$95$0, If[LessEqual[im$95$m, 1.32e+154], N[(N[(2.0 * N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.75 or 1.31999999999999998e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 83.0

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

   5. Applied rewrites 83.0%

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

   if 1.75 < im < 1.31999999999999998e154

   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 re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      7. lower-*.f64 75.8

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)} \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right) \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right) \]

      7. neg-sub0 N/A

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

      8. cosh-undef N/A

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

      9. lift-cosh.f64 N/A

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

      10. *-lft-identity N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   7. Applied rewrites 75.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right) \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot \sin re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sin re) 5e-6)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (fma im_m im_m 2.0))
   (* (* 0.5 re) (fma im_m im_m 2.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (sin(re) <= 5e-6) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(im_m, im_m, 2.0);
	} else {
		tmp = (0.5 * re) * fma(im_m, im_m, 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (sin(re) <= 5e-6)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * fma(im_m, im_m, 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sin[re], $MachinePrecision], 5e-6], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 5.00000000000000041e-6

   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 72.1

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

   5. Applied rewrites 72.1%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 57.4

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

   8. Applied rewrites 57.4%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       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. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 57.4

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

   11. Applied rewrites 57.4%

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

   if 5.00000000000000041e-6 < (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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 75.0

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

   5. Applied rewrites 75.0%

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

   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. unpow2 N/A

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

      6. cube-mult N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 17.1

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

   8. Applied rewrites 17.1%

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

   9. Taylor expanded in re around 0

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

       1. lower-*.f64 22.5

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

   11. Applied rewrites 22.5%

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

Alternative 13: 47.9% accurate, 18.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* (* 0.5 re) (fma im_m im_m 2.0)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return (0.5 * re) * fma(im_m, im_m, 2.0);
}

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(Float64(0.5 * re) * fma(im_m, im_m, 2.0))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

   \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
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 72.8

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

5. Applied rewrites 72.8%

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

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. unpow2 N/A

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

   6. cube-mult N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-pow.f64 47.3

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

8. Applied rewrites 47.3%

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

9. Taylor expanded in re around 0

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

    1. lower-*.f64 45.2

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

11. Applied rewrites 45.2%

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

Alternative 14: 26.7% accurate, 28.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 2 \cdot \left(0.5 \cdot re\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* 2.0 (* 0.5 re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 2.0 * (0.5 * re);
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 2.0d0 * (0.5d0 * re)
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 2.0 * (0.5 * re);
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 2.0 * (0.5 * re)

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(2.0 * Float64(0.5 * re))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 2.0 * (0.5 * re);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(2.0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. 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 72.8

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

5. Applied rewrites 72.8%

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

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. unpow2 N/A

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

   6. cube-mult N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-pow.f64 47.3

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

8. Applied rewrites 47.3%

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

9. Taylor expanded in re around 0

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

    1. lower-*.f64 45.2

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

11. Applied rewrites 45.2%

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

12. Taylor expanded in im around 0

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

14. Applied rewrites 24.3%

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

15. Final simplification 24.3%

    \[\leadsto 2 \cdot \left(0.5 \cdot re\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))