math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 20

Speedup: 1.5×

• 48.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub0-neg N/A

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

   15. lower-neg.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-*l* N/A

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

   20. lower-*.f64 N/A

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

   21. lower-*.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 72.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 88.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 78.2

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

   8. Simplified 78.2%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({re}^{3} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {re}^{3}\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot {re}^{3}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot {re}^{3}\right)} \]

   11. Simplified 28.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   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. Taylor expanded in im around 0

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

   5. Simplified 76.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 67.5

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

   8. Simplified 67.5%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 54.7%

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

   12. Taylor expanded in im around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. lower-fma.f64 N/A

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

       14. lower-*.f64 54.7

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

   14. Simplified 54.7%

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

4. Final simplification 72.8%

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

Alternative 3: 71.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 88.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 78.2

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

   8. Simplified 78.2%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({re}^{3} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {re}^{3}\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot {re}^{3}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot {re}^{3}\right)} \]

   11. Simplified 28.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 99.7

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

   5. Simplified 99.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 76.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 67.5

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

   8. Simplified 67.5%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 54.7%

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

   12. Taylor expanded in im around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. lower-fma.f64 N/A

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

       14. lower-*.f64 54.7

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

   14. Simplified 54.7%

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

4. Final simplification 72.7%

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

Alternative 4: 56.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Simplified 91.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 65.3

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

   8. Simplified 65.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 84.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 44.6

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

   8. Simplified 44.6%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 36.7%

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

   12. Taylor expanded in im around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. lower-fma.f64 N/A

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

       14. lower-*.f64 36.7

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

   14. Simplified 36.7%

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

4. Final simplification 54.7%

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

Alternative 5: 52.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.4

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

   5. Simplified 80.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 58.1

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

   8. Simplified 58.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 84.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 44.6

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

   8. Simplified 44.6%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 36.7%

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

   12. Taylor expanded in im around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. lower-fma.f64 N/A

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

       14. lower-*.f64 36.7

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

   14. Simplified 36.7%

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

4. Final simplification 50.2%

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

Alternative 6: 52.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.4

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

   5. Simplified 80.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 58.1

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

   8. Simplified 58.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 84.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 44.6

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

   8. Simplified 44.6%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 36.7%

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

   12. Taylor expanded in im around inf

       \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{4}\right)}, re\right) \]
   13. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 36.7

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

   14. Simplified 36.7%

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

4. Final simplification 50.2%

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

Alternative 7: 51.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 80.4

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

   5. Simplified 80.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 58.1

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

   8. Simplified 58.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Simplified 84.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 29.7

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

   8. Simplified 29.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 36.7

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

   10. Applied egg-rr 36.7%

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

4. Final simplification 50.2%

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

Alternative 8: 44.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 71.3

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

   7. Simplified 71.3%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3} + re} \]

      2. lower-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.2

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

   10. Simplified 45.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Simplified 84.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 29.7

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

   8. Simplified 29.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 36.7

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

   10. Applied egg-rr 36.7%

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

4. Final simplification 42.0%

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

Alternative 9: 42.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 71.3

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

   7. Simplified 71.3%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3} + re} \]

      2. lower-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.2

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

   10. Simplified 45.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Simplified 84.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 29.7

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

   8. Simplified 29.7%

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

4. Final simplification 39.5%

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

Alternative 10: 42.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 71.3

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

   7. Simplified 71.3%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3} + re} \]

      2. lower-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.2

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

   10. Simplified 45.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Simplified 84.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 29.7

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

   8. Simplified 29.7%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 29.7

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

   11. Simplified 29.7%

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

4. Final simplification 39.5%

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

Alternative 11: 42.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 71.3

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

   7. Simplified 71.3%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3} + re} \]

      2. lower-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.2

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

   10. Simplified 45.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Simplified 84.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 29.7

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

   8. Simplified 29.7%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 29.2

           \[\leadsto im \cdot \left(im \cdot \left(0.041666666666666664 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot re\right)\right)\right) \]

   11. Simplified 29.2%

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

4. Final simplification 39.3%

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

Alternative 12: 40.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 71.3

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Simplified 71.3%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   8. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re + \frac{-1}{6} \cdot {re}^{3}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3} + re} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {re}^{3}, re\right)} \]

      3. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot \color{blue}{{re}^{2}}, re\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{re \cdot {re}^{2}}, re\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

      7. lower-*.f64 45.2

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

   10. Simplified 45.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)} \]

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \]

      9. *-commutative N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      14. lower-*.f64 64.6

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 64.6%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2}, re\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{2} \cdot {im}^{2}}, re\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

      7. lower-*.f64 21.7

         \[\leadsto \mathsf{fma}\left(re, 0.5 \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

   8. Simplified 21.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, 0.5 \cdot \left(im \cdot im\right), re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, 0.5 \cdot \left(im \cdot im\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 0.0001)
   (fma -0.16666666666666666 (* re (* re re)) re)
   (* (* im im) (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 0.0001) {
		tmp = fma(-0.16666666666666666, (re * (re * re)), re);
	} else {
		tmp = (im * im) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0001)
		tmp = fma(-0.16666666666666666, Float64(re * Float64(re * re)), re);
	else
		tmp = Float64(Float64(im * im) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0001], N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 71.3

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Simplified 71.3%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   8. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re + \frac{-1}{6} \cdot {re}^{3}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3} + re} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {re}^{3}, re\right)} \]

      3. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot \color{blue}{{re}^{2}}, re\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{re \cdot {re}^{2}}, re\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

      7. lower-*.f64 45.2

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

   10. Simplified 45.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)} \]

   if 1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \]

      9. *-commutative N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      14. lower-*.f64 64.6

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 64.6%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{re} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2}, re\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{2} \cdot {im}^{2}}, re\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

      7. lower-*.f64 21.7

         \[\leadsto \mathsf{fma}\left(re, 0.5 \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

   8. Simplified 21.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, 0.5 \cdot \left(im \cdot im\right), re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \frac{1}{2}\right)} \cdot re \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)} \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{2} \cdot re\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{2} \cdot re\right) \]

       7. lower-*.f64 21.3

          \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(0.5 \cdot re\right)} \]

   11. Simplified 21.3%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 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-sin.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. lift--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

   4. lift-exp.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

   5. lift-exp.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

   6. lift-+.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

5. Final simplification 100.0%

   \[\leadsto \sin re \cdot \cosh im \]
6. Add Preprocessing

Alternative 15: 87.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{if}\;im \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot t\_0\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma 0.5 (* im im) 1.0)))
   (if (<= im 3.4e-5)
     (* (sin re) t_0)
     (if (<= im 7e+51)
       (* (cosh im) (fma -0.16666666666666666 (* re (* re re)) re))
       (*
        (sin re)
        (fma
         (fma im (* im 0.001388888888888889) 0.041666666666666664)
         (* im (* im (* im im)))
         t_0))))))

C

double code(double re, double im) {
	double t_0 = fma(0.5, (im * im), 1.0);
	double tmp;
	if (im <= 3.4e-5) {
		tmp = sin(re) * t_0;
	} else if (im <= 7e+51) {
		tmp = cosh(im) * fma(-0.16666666666666666, (re * (re * re)), re);
	} else {
		tmp = sin(re) * fma(fma(im, (im * 0.001388888888888889), 0.041666666666666664), (im * (im * (im * im))), t_0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(0.5, Float64(im * im), 1.0)
	tmp = 0.0
	if (im <= 3.4e-5)
		tmp = Float64(sin(re) * t_0);
	elseif (im <= 7e+51)
		tmp = Float64(cosh(im) * fma(-0.16666666666666666, Float64(re * Float64(re * re)), re));
	else
		tmp = Float64(sin(re) * fma(fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), Float64(im * Float64(im * Float64(im * im))), t_0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[im, 3.4e-5], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[im, 7e+51], N[(N[Cosh[im], $MachinePrecision] * N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.4e-5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \]

      9. *-commutative N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      14. lower-*.f64 84.7

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 84.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   if 3.4e-5 < im < 7e51

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   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

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 83.2

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Simplified 83.2%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]
   8. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) + re\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right)} \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) + re\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}\right) + re\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)} + re\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot \frac{-1}{6}, re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot \frac{-1}{6}, re\right) \cdot \left(1 \cdot \cosh im\right)} \]

      7. lower-*.f64 83.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \left(1 \cdot \cosh im\right)} \]

      8. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) + re\right)} \cdot \left(1 \cdot \cosh im\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \frac{-1}{6}} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \frac{-1}{6} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      12. cube-unmult N/A

          \[\leadsto \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{\frac{-1}{6} \cdot {re}^{3}} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {re}^{3}, re\right)} \cdot \left(1 \cdot \cosh im\right) \]

      15. cube-unmult N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \cdot \left(1 \cdot \cosh im\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \cdot \left(1 \cdot \cosh im\right) \]

      17. lower-*.f64 83.2

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \cdot \left(1 \cdot \cosh im\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\left(1 \cdot \cosh im\right)} \]

      19. *-lft-identity 83.2

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\cosh im} \]

   9. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \cosh im} \]

   if 7e51 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 0.0001:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 0.0001)
   (*
    re
    (*
     (fma (* re re) -0.16666666666666666 1.0)
     (fma
      im
      (*
       im
       (fma
        (* im im)
        (fma (* im im) 0.001388888888888889 0.041666666666666664)
        0.5))
      1.0)))
   (fma re (* (* im im) (* (* im im) (* (* im im) 0.001388888888888889))) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 0.0001) {
		tmp = re * (fma((re * re), -0.16666666666666666, 1.0) * fma(im, (im * fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0));
	} else {
		tmp = fma(re, ((im * im) * ((im * im) * ((im * im) * 0.001388888888888889))), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 0.0001)
		tmp = Float64(re * Float64(fma(Float64(re * re), -0.16666666666666666, 1.0) * fma(im, Float64(im * fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5)), 1.0)));
	else
		tmp = fma(re, Float64(Float64(im * im) * Float64(Float64(im * im) * Float64(Float64(im * im) * 0.001388888888888889))), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 0.0001], N[(re * N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1.00000000000000005e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

   5. Simplified 92.9%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 69.7%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)} \]

   if 1.00000000000000005e-4 < (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

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

   5. Simplified 88.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      8. lower-*.f64 29.5

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   8. Simplified 29.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   11. Simplified 18.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \color{blue}{\left(\frac{1}{720} \cdot {im}^{4}\right)}, re\right) \]
   13. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(\frac{1}{720} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right), re\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(\frac{1}{720} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right), re\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \color{blue}{\left(\left(\frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}\right)}, re\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)}, re\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right)}, re\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right), re\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{720} \cdot {im}^{2}\right)\right), re\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{720}\right)}\right), re\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{720}\right)}\right), re\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{720}\right)\right), re\right) \]

       11. lower-*.f64 18.0

           \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.001388888888888889\right)\right), re\right) \]

   14. Simplified 18.0%

       \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)}, re\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 57.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.01)
   (*
    (fma
     (* im im)
     (fma
      im
      (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))
      0.5)
     1.0)
    (* re (* -0.16666666666666666 (* re re))))
   (fma
    re
    (*
     im
     (*
      im
      (fma
       (* im im)
       (fma im (* im 0.001388888888888889) 0.041666666666666664)
       0.5)))
    re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = fma((im * im), fma(im, (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) * (re * (-0.16666666666666666 * (re * re)));
	} else {
		tmp = fma(re, (im * (im * fma((im * im), fma(im, (im * 0.001388888888888889), 0.041666666666666664), 0.5))), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) * Float64(re * Float64(-0.16666666666666666 * Float64(re * re))));
	else
		tmp = fma(re, Float64(im * Float64(im * fma(Float64(im * im), fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), 0.5))), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $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

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

   5. Simplified 97.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      8. lower-*.f64 28.3

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   8. Simplified 28.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({re}^{3} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {re}^{3}\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot {re}^{3}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{-1}{6} \cdot {re}^{3}\right)} \]

   11. Simplified 28.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right)} \]

   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

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]

   5. Simplified 90.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      8. lower-*.f64 69.7

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   8. Simplified 69.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + re \cdot 1} \]

       3. *-rgt-identity N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \color{blue}{re} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), re\right)} \]

   11. Simplified 66.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)} \]

   12. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(re, \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}, re\right) \]
   13. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right), re\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}, re\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}, re\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}, re\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, im \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}\right)}\right), re\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, im \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right)}\right), re\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right)\right), re\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right)\right), re\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, \frac{1}{2}\right)\right), re\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right)\right), re\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{720} + \frac{1}{24}, \frac{1}{2}\right)\right), re\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \frac{1}{720}\right)} + \frac{1}{24}, \frac{1}{2}\right)\right), re\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right)\right), re\right) \]

       14. lower-*.f64 66.1

           \[\leadsto \mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, \color{blue}{im \cdot 0.001388888888888889}, 0.041666666666666664\right), 0.5\right)\right), re\right) \]

   14. Simplified 66.1%

       \[\leadsto \mathsf{fma}\left(re, \color{blue}{im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right)}, re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 86.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.4e-5)
   (* (sin re) (fma 0.5 (* im im) 1.0))
   (if (<= im 2.5e+77)
     (* (cosh im) (fma -0.16666666666666666 (* re (* re re)) re))
     (*
      (sin re)
      (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.4e-5) {
		tmp = sin(re) * fma(0.5, (im * im), 1.0);
	} else if (im <= 2.5e+77) {
		tmp = cosh(im) * fma(-0.16666666666666666, (re * (re * re)), re);
	} else {
		tmp = sin(re) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.4e-5)
		tmp = Float64(sin(re) * fma(0.5, Float64(im * im), 1.0));
	elseif (im <= 2.5e+77)
		tmp = Float64(cosh(im) * fma(-0.16666666666666666, Float64(re * Float64(re * re)), re));
	else
		tmp = Float64(sin(re) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.4e-5], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+77], N[(N[Cosh[im], $MachinePrecision] * N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.4e-5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \]

      9. *-commutative N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      14. lower-*.f64 84.7

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 84.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   if 3.4e-5 < im < 2.50000000000000002e77

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   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

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 71.3

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Simplified 71.3%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]
   8. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) + re\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right)} \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) + re\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}\right) + re\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)} + re\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot \frac{-1}{6}, re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot \frac{-1}{6}, re\right) \cdot \left(1 \cdot \cosh im\right)} \]

      7. lower-*.f64 71.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \left(1 \cdot \cosh im\right)} \]

      8. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) + re\right)} \cdot \left(1 \cdot \cosh im\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \frac{-1}{6}} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \frac{-1}{6} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      12. cube-unmult N/A

          \[\leadsto \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{\frac{-1}{6} \cdot {re}^{3}} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {re}^{3}, re\right)} \cdot \left(1 \cdot \cosh im\right) \]

      15. cube-unmult N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \cdot \left(1 \cdot \cosh im\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \cdot \left(1 \cdot \cosh im\right) \]

      17. lower-*.f64 71.3

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \cdot \left(1 \cdot \cosh im\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\left(1 \cdot \cosh im\right)} \]

      19. *-lft-identity 71.3

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\cosh im} \]

   9. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \cosh im} \]

   if 2.50000000000000002e77 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right)\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right)} \]

      4. associate-*r* N/A

         \[\leadsto {im}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left({im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re\right) + \sin re\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left({im}^{2} \cdot \frac{1}{2}\right) \cdot \sin re} + \sin re\right) \]

      7. unpow2 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{2}\right) \cdot \sin re + \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(im \cdot \left(im \cdot \frac{1}{2}\right)\right)} \cdot \sin re + \sin re\right) \]

      9. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\left(im \cdot \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \cdot \sin re + \sin re\right) \]

      10. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \cdot \sin re} \]

      11. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 84.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{if}\;im \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) (fma 0.5 (* im im) 1.0))))
   (if (<= im 3.4e-5)
     t_0
     (if (<= im 1.35e+154)
       (* (cosh im) (fma -0.16666666666666666 (* re (* re re)) re))
       t_0))))

C

double code(double re, double im) {
	double t_0 = sin(re) * fma(0.5, (im * im), 1.0);
	double tmp;
	if (im <= 3.4e-5) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = cosh(im) * fma(-0.16666666666666666, (re * (re * re)), re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * fma(0.5, Float64(im * im), 1.0))
	tmp = 0.0
	if (im <= 3.4e-5)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = Float64(cosh(im) * fma(-0.16666666666666666, Float64(re * Float64(re * re)), re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3.4e-5], t$95$0, If[LessEqual[im, 1.35e+154], N[(N[Cosh[im], $MachinePrecision] * N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 3.4e-5 or 1.35000000000000003e154 < 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

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \]

      9. *-commutative N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      14. lower-*.f64 86.3

          \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

   5. Simplified 86.3%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

   if 3.4e-5 < im < 1.35000000000000003e154

   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-sin.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\sin re}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{\color{blue}{0 - im}} + e^{im}\right) \]

      4. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{e^{0 - im}} + e^{im}\right) \]

      5. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{e^{im}}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \sin re\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \sin re} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. lower-*.f64 76.3

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   7. Simplified 76.3%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]
   8. Step-by-step derivation

      1. lift-cosh.f64 N/A

         \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) + re\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right)} \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) + re\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}\right) + re\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)} + re\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot \frac{-1}{6}, re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot \frac{-1}{6}, re\right) \cdot \left(1 \cdot \cosh im\right)} \]

      7. lower-*.f64 76.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \left(1 \cdot \cosh im\right)} \]

      8. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) + re\right)} \cdot \left(1 \cdot \cosh im\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right)} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \frac{-1}{6}} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \frac{-1}{6} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      12. cube-unmult N/A

          \[\leadsto \left(\color{blue}{{re}^{3}} \cdot \frac{-1}{6} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\color{blue}{\frac{-1}{6} \cdot {re}^{3}} + re\right) \cdot \left(1 \cdot \cosh im\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {re}^{3}, re\right)} \cdot \left(1 \cdot \cosh im\right) \]

      15. cube-unmult N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \cdot \left(1 \cdot \cosh im\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \cdot \left(1 \cdot \cosh im\right) \]

      17. lower-*.f64 76.3

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \cdot \left(1 \cdot \cosh im\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\left(1 \cdot \cosh im\right)} \]

      19. *-lft-identity 76.3

          \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\cosh im} \]

   9. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \cosh im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 24.3% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \left(im \cdot im\right) \cdot \left(re \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* im im) (* re 0.5)))

C

double code(double re, double im) {
	return (im * im) * (re * 0.5);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (im * im) * (re * 0.5d0)
end function

Java

public static double code(double re, double im) {
	return (im * im) * (re * 0.5);
}

Python

def code(re, im):
	return (im * im) * (re * 0.5)

Julia

function code(re, im)
	return Float64(Float64(im * im) * Float64(re * 0.5))
end

MATLAB

function tmp = code(re, im)
	tmp = (im * im) * (re * 0.5);
end

Wolfram

code[re_, im_] := N[(N[(im * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

   2. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   3. unpow2 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

   4. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

   5. *-commutative N/A

      \[\leadsto \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

   8. lower-sin.f64 N/A

      \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \]

   9. *-commutative N/A

      \[\leadsto \sin re \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \]

   10. associate-*r* N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(im \cdot im\right)} + 1\right) \]

   11. unpow2 N/A

       \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{{im}^{2}} + 1\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \]

   13. unpow2 N/A

       \[\leadsto \sin re \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

   14. lower-*.f64 74.5

       \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \]

5. Simplified 74.5%

   \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + re \cdot 1} \]

   3. *-rgt-identity N/A

      \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{re} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2}, re\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{2} \cdot {im}^{2}}, re\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(re, \frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

   7. lower-*.f64 40.4

      \[\leadsto \mathsf{fma}\left(re, 0.5 \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

8. Simplified 40.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(re, 0.5 \cdot \left(im \cdot im\right), re\right)} \]

9. Taylor expanded in im around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{\left({im}^{2} \cdot \frac{1}{2}\right)} \cdot re \]

    3. associate-*r* N/A

       \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} \cdot re\right)} \]

    5. unpow2 N/A

       \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{2} \cdot re\right) \]

    6. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{2} \cdot re\right) \]

    7. lower-*.f64 17.4

       \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(0.5 \cdot re\right)} \]

11. Simplified 17.4%

    \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)} \]

12. Final simplification 17.4%

    \[\leadsto \left(im \cdot im\right) \cdot \left(re \cdot 0.5\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))