math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 20

Speedup: 1.5×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub0-neg N/A

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

   13. cosh-undef N/A

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

   14. associate-*r* N/A

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

   15. metadata-eval N/A

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

   16. exp-0 N/A

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

   17. lower-*.f64 N/A

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

   18. exp-0 N/A

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

   19. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 81.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * re), $MachinePrecision] * N[Cosh[im], $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 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 48.6

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

   8. Applied rewrites 48.6%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

      1. 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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(1 \cdot re\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.4%

       \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(1 \cdot re\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-lft-identity 78.4

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

   12. Applied rewrites 78.4%

       \[\leadsto \color{blue}{\cosh im} \cdot \left(1 \cdot re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

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

Alternative 3: 80.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(im * im + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(1.0 * re), $MachinePrecision] * N[Cosh[im], $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 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 48.6

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

   8. Applied rewrites 48.6%

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

   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 99.8

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

   5. Applied rewrites 99.8%

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

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

   1. Initial program 100.0%

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

      1. 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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(1 \cdot re\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.4%

       \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(1 \cdot re\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-lft-identity 78.4

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

   12. Applied rewrites 78.4%

       \[\leadsto \color{blue}{\cosh im} \cdot \left(1 \cdot re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

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

Alternative 4: 80.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(1.0 * re), $MachinePrecision] * N[Cosh[im], $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 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 48.6

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

   8. Applied rewrites 48.6%

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

   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

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

      1. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

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

   1. Initial program 100.0%

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

      1. 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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(1 \cdot re\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.4%

       \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(1 \cdot re\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-lft-identity 78.4

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

   12. Applied rewrites 78.4%

       \[\leadsto \color{blue}{\cosh im} \cdot \left(1 \cdot re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

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

Alternative 5: 78.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp im) (exp (- im))) (* 0.5 (sin re)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma im im 2.0)
      (*
       (fma
        (fma
         (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
         (* re re)
         -0.08333333333333333)
        (* re re)
        0.5)
       re))
     (if (<= t_0 1.0)
       (sin re)
       (*
        (* 0.5 re)
        (fma
         (fma
          (fma 0.002777777777777778 (* im im) 0.08333333333333333)
          (* im im)
          1.0)
         (* im im)
         2.0))))))

C

double code(double re, double im) {
	double t_0 = (exp(im) + exp(-im)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(im, im, 2.0) * (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(im, im, 2.0) * Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 48.6

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

   8. Applied rewrites 48.6%

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

   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

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

      1. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 74.7

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

   8. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

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

Alternative 6: 89.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

   5. Applied rewrites 96.3%

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

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

   1. Initial program 100.0%

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

      1. 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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(1 \cdot re\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.4%

       \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(1 \cdot re\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-lft-identity 78.4

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

   12. Applied rewrites 78.4%

       \[\leadsto \color{blue}{\cosh im} \cdot \left(1 \cdot re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

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

Alternative 7: 89.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

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

   1. Initial program 100.0%

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

      1. 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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(1 \cdot re\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.4%

       \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(1 \cdot re\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-lft-identity 78.4

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

   12. Applied rewrites 78.4%

       \[\leadsto \color{blue}{\cosh im} \cdot \left(1 \cdot re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

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

Alternative 8: 53.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) 0.29)
   (*
    (fma im im 2.0)
    (*
     (fma
      (fma
       (fma -9.92063492063492e-5 (* re re) 0.004166666666666667)
       (* re re)
       -0.08333333333333333)
      (* re re)
      0.5)
     re))
   (*
    (* 0.5 re)
    (fma
     (fma
      (fma 0.002777777777777778 (* im im) 0.08333333333333333)
      (* im im)
      1.0)
     (* im im)
     2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= 0.29) {
		tmp = fma(im, im, 2.0) * (fma(fma(fma(-9.92063492063492e-5, (re * re), 0.004166666666666667), (re * re), -0.08333333333333333), (re * re), 0.5) * re);
	} else {
		tmp = (0.5 * re) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= 0.29)
		tmp = Float64(fma(im, im, 2.0) * Float64(fma(fma(fma(-9.92063492063492e-5, Float64(re * re), 0.004166666666666667), Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re));
	else
		tmp = Float64(Float64(0.5 * re) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.29], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(N[(-9.92063492063492e-5 * N[(re * re), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.28999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 65.1

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

   8. Applied rewrites 65.1%

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

   if 0.28999999999999998 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 53.0

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

   8. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right), re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) 0.29)
   (*
    (* (fma -0.16666666666666666 (* re re) 1.0) re)
    (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0))
   (*
    (* 0.5 re)
    (fma
     (fma
      (fma 0.002777777777777778 (* im im) 0.08333333333333333)
      (* im im)
      1.0)
     (* im im)
     2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= 0.29) {
		tmp = (fma(-0.16666666666666666, (re * re), 1.0) * re) * fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0);
	} else {
		tmp = (0.5 * re) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= 0.29)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re) * fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0));
	else
		tmp = Float64(Float64(0.5 * re) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.29], N[(N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.28999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.2%

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

   if 0.28999999999999998 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 53.0

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

   8. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) 0.29)
   (*
    (* (fma -0.16666666666666666 (* re re) 1.0) re)
    (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0))
   (* (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= 0.29) {
		tmp = (fma(-0.16666666666666666, (re * re), 1.0) * re) * fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0);
	} else {
		tmp = fma((0.041666666666666664 * (im * im)), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= 0.29)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re) * fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0));
	else
		tmp = Float64(fma(Float64(0.041666666666666664 * Float64(im * im)), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.29], N[(N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.28999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.2%

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

   if 0.28999999999999998 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 49.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 49.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) 0.29)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (fma im im 2.0))
   (* (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= 0.29) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = fma((0.041666666666666664 * (im * im)), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= 0.29)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(Float64(0.041666666666666664 * Float64(im * im)), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.29], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.28999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in re around 0

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

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 64.4

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

   8. Applied rewrites 64.4%

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

   if 0.28999999999999998 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 49.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 49.2%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) 0.29)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= 0.29) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = fma((0.041666666666666664 * (im * im)), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= 0.29)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(fma(Float64(0.041666666666666664 * Float64(im * im)), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.29], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.28999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.0%

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

   if 0.28999999999999998 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 49.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 49.2%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.041666666666666664 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) 0.29)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (* (* (* 0.041666666666666664 (* im im)) im) im) re)))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= 0.29) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = (((0.041666666666666664 * (im * im)) * im) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= 0.29)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(Float64(Float64(Float64(0.041666666666666664 * Float64(im * im)) * im) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.29], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.28999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.0%

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

   if 0.28999999999999998 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 49.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 49.2%

       \[\leadsto \left(\left(\left(0.041666666666666664 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.041666666666666664 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.041666666666666664\right) \cdot im\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) 0.29)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (* (* (* (* im im) re) 0.041666666666666664) im) im)))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= 0.29) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = ((((im * im) * re) * 0.041666666666666664) * im) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= 0.29)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(im * im) * re) * 0.041666666666666664) * im) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.29], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * re), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * im), $MachinePrecision] * im), $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))) < 0.28999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.0%

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

   if 0.28999999999999998 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 49.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 45.4%

       \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.041666666666666664\right) \cdot im\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0.29:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.041666666666666664\right) \cdot im\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) 2e-309)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (* 0.5 re) (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= 2e-309) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= 2e-309)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-309], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.9999999999999988e-309

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.6%

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

   if 1.9999999999999988e-309 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.1

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

   8. Applied rewrites 58.1%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -0.01:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;1 \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* 0.5 (sin re))) -0.01)
   (* (* (* re re) re) -0.16666666666666666)
   (* 1.0 re)))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= -0.01) {
		tmp = ((re * re) * re) * -0.16666666666666666;
	} else {
		tmp = 1.0 * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((exp(im) + exp(-im)) * (0.5d0 * sin(re))) <= (-0.01d0)) then
        tmp = ((re * re) * re) * (-0.16666666666666666d0)
    else
        tmp = 1.0d0 * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((Math.exp(im) + Math.exp(-im)) * (0.5 * Math.sin(re))) <= -0.01) {
		tmp = ((re * re) * re) * -0.16666666666666666;
	} else {
		tmp = 1.0 * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((math.exp(im) + math.exp(-im)) * (0.5 * math.sin(re))) <= -0.01:
		tmp = ((re * re) * re) * -0.16666666666666666
	else:
		tmp = 1.0 * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * sin(re))) <= -0.01)
		tmp = Float64(Float64(Float64(re * re) * re) * -0.16666666666666666);
	else
		tmp = Float64(1.0 * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((exp(im) + exp(-im)) * (0.5 * sin(re))) <= -0.01)
		tmp = ((re * re) * re) * -0.16666666666666666;
	else
		tmp = 1.0 * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(1.0 * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 32.6

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

   5. Applied rewrites 32.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 12.4%

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

   9. Taylor expanded in re around inf

      \[\leadsto \frac{-1}{6} \cdot {re}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.7%

       \[\leadsto \left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666 \]

   if -0.0100000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 59.2%

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

   9. Taylor expanded in re around 0

      \[\leadsto 1 \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 53.6%

       \[\leadsto 1 \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -0.01:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;1 \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.08333333333333333\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.01)
   (* (* (* (* re re) re) -0.08333333333333333) (fma im im 2.0))
   (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = (((re * re) * re) * -0.08333333333333333) * fma(im, im, 2.0);
	} else {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(re * re) * re) * -0.08333333333333333) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in re around 0

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

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 21.1

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

   8. Applied rewrites 21.1%

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

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{-1}{12} \cdot \color{blue}{{re}^{3}}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 20.4%

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

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 74.4%

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

Alternative 18: 54.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot -0.08333333333333333\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.01)
   (* (* (* (* re re) re) -0.08333333333333333) (fma im im 2.0))
   (* (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = (((re * re) * re) * -0.08333333333333333) * fma(im, im, 2.0);
	} else {
		tmp = fma((0.041666666666666664 * (im * im)), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(re * re) * re) * -0.08333333333333333) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(Float64(0.041666666666666664 * Float64(im * im)), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in re around 0

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

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 21.1

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

   8. Applied rewrites 21.1%

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

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{-1}{12} \cdot \color{blue}{{re}^{3}}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 20.4%

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

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 74.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 74.2%

       \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 33.8% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (fma -0.16666666666666666 (* re re) 1.0) re))

C

double code(double re, double im) {
	return fma(-0.16666666666666666, (re * re), 1.0) * re;
}

Julia

function code(re, im)
	return Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re)
end

Wolfram

code[re_, im_] := N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-sin.f64 57.0

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

5. Applied rewrites 57.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 41.1%

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

Alternative 20: 26.1% accurate, 52.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 1.0 re))

C

double code(double re, double im) {
	return 1.0 * re;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 * re
end function

Java

public static double code(double re, double im) {
	return 1.0 * re;
}

Python

def code(re, im):
	return 1.0 * re

Julia

function code(re, im)
	return Float64(1.0 * re)
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 * re;
end

Wolfram

code[re_, im_] := N[(1.0 * re), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-sin.f64 57.0

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

5. Applied rewrites 57.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 42.6%

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

9. Taylor expanded in re around 0

   \[\leadsto 1 \cdot re \]
10. Step-by-step derivation

11. Applied rewrites 35.9%

    \[\leadsto 1 \cdot re \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))