math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 17

Speedup: 1.5×

• 49.0% 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

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

FPCore

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

C

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

Julia

im_m = abs(im)
function code(re, im_m)
	return fma(Float64(0.5 / exp(im_m)), sin(re), 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[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[Sin[re], $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-rgt-in N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift--.f64 N/A

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

   9. exp-diff N/A

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

   10. lift-exp.f64 N/A

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

   11. associate-*l/ N/A

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

   12. exp-0 N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{e^{im}}, \sin re, \left(e^{im} \cdot 0.5\right) \cdot \sin re\right)} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} + e^{im\_m}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im\_m \cdot im\_m, 0.041666666666666664\right), im\_m \cdot im\_m, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left({im\_m}^{4}, \mathsf{fma}\left(0.002777777777777778, im\_m \cdot im\_m, 0.08333333333333333\right), \mathsf{fma}\left(im\_m, im\_m, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im_m)) (exp im_m)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
         (* re re)
         -0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma im_m im_m 2.0))
     (if (<= t_0 1.0)
       (*
        (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))
       (*
        (* 0.5 re)
        (fma
         (pow im_m 4.0)
         (fma 0.002777777777777778 (* im_m im_m) 0.08333333333333333)
         (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)) * (exp(-im_m) + exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	} else if (t_0 <= 1.0) {
		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);
	} else {
		tmp = (0.5 * re) * fma(pow(im_m, 4.0), fma(0.002777777777777778, (im_m * im_m), 0.08333333333333333), fma(im_m, im_m, 2.0));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) + exp(im_m)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	elseif (t_0 <= 1.0)
		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));
	else
		tmp = Float64(Float64(0.5 * re) * fma((im_m ^ 4.0), fma(0.002777777777777778, Float64(im_m * im_m), 0.08333333333333333), 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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], 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], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Power[im$95$m, 4.0], $MachinePrecision] * N[(0.002777777777777778 * N[(im$95$m * im$95$m), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $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 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. 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 47.6

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

   8. Applied rewrites 47.6%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   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 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 2.3

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

   8. Applied rewrites 2.3%

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

   9. Taylor expanded in im around 0

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

       1. distribute-lft-in N/A

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

       2. *-rgt-identity N/A

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

       3. associate-+r+ N/A

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

       4. +-commutative N/A

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

       5. associate-*r* N/A

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

       6. pow-sqr N/A

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

       7. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{\color{blue}{4}} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + \left(2 + {im}^{2}\right)\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, \frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, 2 + {im}^{2}\right)} \]

       9. lower-pow.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{4}}, \frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, 2 + {im}^{2}\right) \]

       10. +-commutative N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, 2 + {im}^{2}\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, 2 + {im}^{2}\right) \]

       12. unpow2 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), 2 + {im}^{2}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), 2 + {im}^{2}\right) \]

       14. +-commutative N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{{im}^{2} + 2}\right) \]

       15. unpow2 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im} + 2\right) \]

       16. lower-fma.f64 62.6

           \[\leadsto \left(0.5 \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), \color{blue}{\mathsf{fma}\left(im, im, 2\right)}\right) \]

   11. Applied rewrites 62.6%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), \mathsf{fma}\left(im, im, 2\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), \mathsf{fma}\left(im, im, 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} + e^{im\_m}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im\_m \cdot im\_m, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left({im\_m}^{4}, \mathsf{fma}\left(0.002777777777777778, im\_m \cdot im\_m, 0.08333333333333333\right), \mathsf{fma}\left(im\_m, im\_m, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im_m)) (exp im_m)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
         (* re re)
         -0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma im_m im_m 2.0))
     (if (<= t_0 1.0)
       (*
        (fma (fma 0.041666666666666664 (* im_m im_m) 0.5) (* im_m im_m) 1.0)
        (sin re))
       (*
        (* 0.5 re)
        (fma
         (pow im_m 4.0)
         (fma 0.002777777777777778 (* im_m im_m) 0.08333333333333333)
         (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)) * (exp(-im_m) + exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	} else if (t_0 <= 1.0) {
		tmp = fma(fma(0.041666666666666664, (im_m * im_m), 0.5), (im_m * im_m), 1.0) * sin(re);
	} else {
		tmp = (0.5 * re) * fma(pow(im_m, 4.0), fma(0.002777777777777778, (im_m * im_m), 0.08333333333333333), fma(im_m, im_m, 2.0));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) + exp(im_m)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im_m * im_m), 0.5), Float64(im_m * im_m), 1.0) * sin(re));
	else
		tmp = Float64(Float64(0.5 * re) * fma((im_m ^ 4.0), fma(0.002777777777777778, Float64(im_m * im_m), 0.08333333333333333), 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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.041666666666666664 * 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], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Power[im$95$m, 4.0], $MachinePrecision] * N[(0.002777777777777778 * N[(im$95$m * im$95$m), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $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 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. 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 47.6

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

   8. Applied rewrites 47.6%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \cdot \sin re \]
   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 99.8

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

   7. Applied rewrites 99.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 2.3

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

   8. Applied rewrites 2.3%

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

   9. Taylor expanded in im around 0

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

       1. distribute-lft-in N/A

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

       2. *-rgt-identity N/A

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

       3. associate-+r+ N/A

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

       4. +-commutative N/A

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

       5. associate-*r* N/A

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

       6. pow-sqr N/A

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

       7. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left({im}^{\color{blue}{4}} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + \left(2 + {im}^{2}\right)\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, \frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, 2 + {im}^{2}\right)} \]

       9. lower-pow.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{4}}, \frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, 2 + {im}^{2}\right) \]

       10. +-commutative N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, 2 + {im}^{2}\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, 2 + {im}^{2}\right) \]

       12. unpow2 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), 2 + {im}^{2}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), 2 + {im}^{2}\right) \]

       14. +-commutative N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{{im}^{2} + 2}\right) \]

       15. unpow2 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im} + 2\right) \]

       16. lower-fma.f64 62.6

           \[\leadsto \left(0.5 \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), \color{blue}{\mathsf{fma}\left(im, im, 2\right)}\right) \]

   11. Applied rewrites 62.6%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), \mathsf{fma}\left(im, im, 2\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), \mathsf{fma}\left(im, im, 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 0.4× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im_m)) (exp im_m)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
         (* re re)
         -0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma im_m im_m 2.0))
     (if (<= t_0 1.0)
       (*
        (fma (fma 0.041666666666666664 (* im_m im_m) 0.5) (* im_m im_m) 1.0)
        (sin re))
       (*
        (* 0.5 re)
        (fma
         (* im_m im_m)
         (* (* im_m im_m) 0.08333333333333333)
         (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)) * (exp(-im_m) + exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	} else if (t_0 <= 1.0) {
		tmp = fma(fma(0.041666666666666664, (im_m * im_m), 0.5), (im_m * im_m), 1.0) * sin(re);
	} else {
		tmp = (0.5 * re) * fma((im_m * im_m), ((im_m * im_m) * 0.08333333333333333), fma(im_m, im_m, 2.0));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) + exp(im_m)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im_m * im_m), 0.5), Float64(im_m * im_m), 1.0) * sin(re));
	else
		tmp = Float64(Float64(0.5 * re) * fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * 0.08333333333333333), 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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.041666666666666664 * 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], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $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 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. 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 47.6

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

   8. Applied rewrites 47.6%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \cdot \sin re \]
   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 99.8

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

   7. Applied rewrites 99.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 2.3

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

   8. Applied rewrites 2.3%

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

   9. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   10. 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

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

       6. associate-*r* N/A

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

       7. pow-sqr N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-pow.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 59.5

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

   11. Applied rewrites 59.5%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, 0.08333333333333333, \mathsf{fma}\left(im, im, 2\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 59.5%

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

4. Final simplification 78.0%

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

Alternative 5: 78.0% accurate, 0.4× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* t_0 (+ (exp (- im_m)) (exp im_m)))))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
         (* re re)
         -0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma im_m im_m 2.0))
     (if (<= t_1 1.0)
       (* t_0 (fma im_m im_m 2.0))
       (*
        (* 0.5 re)
        (fma
         (* im_m im_m)
         (* (* im_m im_m) 0.08333333333333333)
         (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 = t_0 * (exp(-im_m) + exp(im_m));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	} else if (t_1 <= 1.0) {
		tmp = t_0 * fma(im_m, im_m, 2.0);
	} else {
		tmp = (0.5 * re) * fma((im_m * im_m), ((im_m * im_m) * 0.08333333333333333), 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(t_0 * Float64(exp(Float64(-im_m)) + exp(im_m)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * 0.08333333333333333), 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[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $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 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. 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 47.6

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

   8. Applied rewrites 47.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 2.3

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

   8. Applied rewrites 2.3%

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

   9. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   10. 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

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

       6. associate-*r* N/A

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

       7. pow-sqr N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-pow.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 59.5

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

   11. Applied rewrites 59.5%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, 0.08333333333333333, \mathsf{fma}\left(im, im, 2\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 59.5%

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

4. Final simplification 77.8%

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

Alternative 6: 77.7% accurate, 0.4× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im_m)) (exp im_m)))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
         (* re re)
         -0.08333333333333333)
        (* re re)
        0.5)
       re)
      (fma im_m im_m 2.0))
     (if (<= t_0 1.0)
       (sin re)
       (*
        (* 0.5 re)
        (fma
         (* im_m im_m)
         (* (* im_m im_m) 0.08333333333333333)
         (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)) * (exp(-im_m) + exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) * fma((im_m * im_m), ((im_m * im_m) * 0.08333333333333333), fma(im_m, im_m, 2.0));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) + exp(im_m)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) * fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * 0.08333333333333333), 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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $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 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. 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 47.6

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

   8. Applied rewrites 47.6%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. 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-rgt-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift--.f64 N/A

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

      9. exp-diff N/A

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

      10. lift-exp.f64 N/A

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

      11. associate-*l/ N/A

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

      12. exp-0 N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-out N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. mul0-rgt N/A

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

      8. mul0-rgt N/A

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

      9. +-rgt-identity N/A

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

      10. lower-sin.f64 98.8

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

   7. Applied rewrites 98.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 2.3

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

   8. Applied rewrites 2.3%

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

   9. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   10. 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

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

       6. associate-*r* N/A

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

       7. pow-sqr N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-pow.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 59.5

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

   11. Applied rewrites 59.5%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, 0.08333333333333333, \mathsf{fma}\left(im, im, 2\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 59.5%

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

4. Final simplification 77.4%

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

Alternative 7: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

im_m = abs(im)
function code(re, im_m)
	return fma(Float64(sin(re) * 0.5), 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[(N[Sin[re], $MachinePrecision] * 0.5), $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-rgt-in N/A

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

   4. *-commutative N/A

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

   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. Add Preprocessing

Alternative 8: 53.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} + e^{im\_m}\right) \leq -0.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \left(im\_m \cdot im\_m\right) \cdot 0.08333333333333333, \mathsf{fma}\left(im\_m, im\_m, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im_m)) (exp im_m))) -0.05)
   (*
    (*
     (fma
      (fma
       (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
       (* re re)
       -0.08333333333333333)
      (* re re)
      0.5)
     re)
    (fma im_m im_m 2.0))
   (*
    (* 0.5 re)
    (fma
     (* im_m im_m)
     (* (* im_m im_m) 0.08333333333333333)
     (fma im_m im_m 2.0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im_m) + exp(im_m))) <= -0.05) {
		tmp = (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re) * fma(im_m, im_m, 2.0);
	} else {
		tmp = (0.5 * re) * fma((im_m * im_m), ((im_m * im_m) * 0.08333333333333333), 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(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) + exp(im_m))) <= -0.05)
		tmp = Float64(Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * 0.08333333333333333), 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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. 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 35.4

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

   8. Applied rewrites 35.4%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 41.4

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

   8. Applied rewrites 41.4%

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

   9. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   10. 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

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

       6. associate-*r* N/A

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

       7. pow-sqr N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-pow.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 62.1

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

   11. Applied rewrites 62.1%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, 0.08333333333333333, \mathsf{fma}\left(im, im, 2\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 62.1%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot 0.08333333333333333, \mathsf{fma}\left(im, im, 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[Sin[re], $MachinePrecision] * N[Cosh[im$95$m], $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-rgt-in N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift--.f64 N/A

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

   9. exp-diff N/A

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

   10. lift-exp.f64 N/A

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

   11. associate-*l/ N/A

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

   12. exp-0 N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-rgt-out N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. 1-exp N/A

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

   10. lift-exp.f64 N/A

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

   11. exp-diff N/A

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

   12. distribute-rgt-in N/A

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

   13. lift-exp.f64 N/A

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

   14. neg-sub0 N/A

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

   15. cosh-undef N/A

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

   16. lift-cosh.f64 N/A

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

   17. *-lft-identity N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*r* N/A

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

   20. metadata-eval N/A

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

   21. *-lft-identity N/A

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

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
7. Add Preprocessing

Alternative 10: 56.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.004:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5} \cdot \left(re \cdot re\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \left(im\_m \cdot im\_m\right) \cdot 0.08333333333333333, \mathsf{fma}\left(im\_m, im\_m, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sin re) -0.004)
   (*
    (*
     (fma
      (fma (* -9.92063492063492e-5 (* re re)) (* re re) -0.08333333333333333)
      (* re re)
      0.5)
     re)
    2.0)
   (*
    (* 0.5 re)
    (fma
     (* im_m im_m)
     (* (* im_m im_m) 0.08333333333333333)
     (fma im_m im_m 2.0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.004) {
		tmp = (fma(fma((-9.92063492063492e-5 * (re * re)), (re * re), -0.08333333333333333), (re * re), 0.5) * re) * 2.0;
	} else {
		tmp = (0.5 * re) * fma((im_m * im_m), ((im_m * im_m) * 0.08333333333333333), 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) <= -0.004)
		tmp = Float64(Float64(fma(fma(Float64(-9.92063492063492e-5 * Float64(re * re)), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * 2.0);
	else
		tmp = Float64(Float64(0.5 * re) * fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * 0.08333333333333333), 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], -0.004], 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] * 2.0), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 22.9

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

   8. Applied rewrites 22.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 22.7%

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

   if -0.0040000000000000001 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 36.2

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

   8. Applied rewrites 36.2%

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

   9. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   10. 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

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

       6. associate-*r* N/A

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

       7. pow-sqr N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-pow.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 65.1

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

   11. Applied rewrites 65.1%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, 0.08333333333333333, \mathsf{fma}\left(im, im, 2\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 65.1%

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

Alternative 11: 56.3% accurate, 2.2× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sin re) -0.004)
   (* (* (* (* re re) -0.08333333333333333) re) (fma im_m im_m 2.0))
   (*
    (* 0.5 re)
    (fma
     (* im_m im_m)
     (* (* im_m im_m) 0.08333333333333333)
     (fma im_m im_m 2.0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.004) {
		tmp = (((re * re) * -0.08333333333333333) * re) * fma(im_m, im_m, 2.0);
	} else {
		tmp = (0.5 * re) * fma((im_m * im_m), ((im_m * im_m) * 0.08333333333333333), 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) <= -0.004)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.08333333333333333) * re) * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * 0.08333333333333333), 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], -0.004], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 26.0

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

   8. Applied rewrites 26.0%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.6%

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

   if -0.0040000000000000001 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 36.2

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

   8. Applied rewrites 36.2%

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

   9. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   10. 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

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

       6. associate-*r* N/A

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

       7. pow-sqr N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-pow.f64 N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. lower-fma.f64 65.1

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

   11. Applied rewrites 65.1%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, 0.08333333333333333, \mathsf{fma}\left(im, im, 2\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 65.1%

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

Alternative 12: 49.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sin re) 2e-9)
   (* (* (fma -0.08333333333333333 (* re re) 0.5) re) (fma im_m im_m 2.0))
   (*
    (*
     (fma
      (fma 0.004166666666666667 (* re re) -0.08333333333333333)
      (* re re)
      0.5)
     re)
    2.0)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (sin(re) <= 2e-9) {
		tmp = (fma(-0.08333333333333333, (re * re), 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) * 2.0;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (sin(re) <= 2e-9)
		tmp = Float64(Float64(fma(-0.08333333333333333, Float64(re * re), 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) * 2.0);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Sin[re], $MachinePrecision], 2e-9], N[(N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 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] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 2.00000000000000012e-9

   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 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 60.2

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

   8. Applied rewrites 60.2%

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

   if 2.00000000000000012e-9 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 18.1

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

   8. Applied rewrites 18.1%

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

Alternative 13: 49.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-213}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sin re) 1e-213)
   (* (* (fma -0.08333333333333333 (* re re) 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) <= 1e-213) {
		tmp = (fma(-0.08333333333333333, (re * re), 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) <= 1e-213)
		tmp = Float64(Float64(fma(-0.08333333333333333, Float64(re * re), 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], 1e-213], N[(N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 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) < 9.9999999999999995e-214

   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 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 56.8

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

   8. Applied rewrites 56.8%

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

   if 9.9999999999999995e-214 < (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 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 35.0

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

   8. Applied rewrites 35.0%

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

Alternative 14: 49.3% accurate, 2.4× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sin re) -0.004)
   (* (* (* (* re re) -0.08333333333333333) 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) <= -0.004) {
		tmp = (((re * re) * -0.08333333333333333) * 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) <= -0.004)
		tmp = Float64(Float64(Float64(Float64(re * re) * -0.08333333333333333) * 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], -0.004], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $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) < -0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 26.0

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

   8. Applied rewrites 26.0%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.6%

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

   if -0.0040000000000000001 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 53.8

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

   8. Applied rewrites 53.8%

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

Alternative 15: 47.8% accurate, 2.5× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sin re) -0.004)
   (* (* (fma -0.08333333333333333 (* re re) 0.5) re) 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) <= -0.004) {
		tmp = (fma(-0.08333333333333333, (re * re), 0.5) * re) * 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) <= -0.004)
		tmp = Float64(Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re) * 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], -0.004], N[(N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * 2.0), $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) < -0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 16.0

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

   8. Applied rewrites 16.0%

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

   if -0.0040000000000000001 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 53.8

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

   8. Applied rewrites 53.8%

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

Alternative 16: 47.7% 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 75.5

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

5. Applied rewrites 75.5%

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

6. Taylor expanded in re around 0

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

   1. lower-*.f64 47.4

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

8. Applied rewrites 47.4%

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

Alternative 17: 27.0% accurate, 28.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Applied rewrites 53.4%

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

6. Taylor expanded in re around 0

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

   1. lower-*.f64 28.9

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

8. Applied rewrites 28.9%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))