math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.2s

Alternatives: 17

Speedup: 1.5×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. sub0-neg N/A

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

   7. cosh-undef N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. exp-0 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. exp-0 N/A

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

   13. cosh-lowering-cosh.f64 N/A

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

   14. sin-lowering-sin.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 86.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (cosh im) (fma re (* (* re re) -0.16666666666666666) re))
     (if (<= t_0 1.0)
       (*
        (sin re)
        (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0))
       (* (cosh im) 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 = cosh(im) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else if (t_0 <= 1.0) {
		tmp = sin(re) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	} else {
		tmp = cosh(im) * 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(cosh(im) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(re) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = Float64(cosh(im) * 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[Cosh[im], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 73.0

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

   7. Simplified 73.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. 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. *-lowering-*.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. sin-lowering-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 99.1%

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

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   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

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

   7. Simplified 80.6%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{re} \]
   8. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 80.6

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

   9. Applied egg-rr 80.6%

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

4. Final simplification 87.1%

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

Alternative 3: 86.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (cosh im) (fma re (* (* re re) -0.16666666666666666) re))
     (if (<= t_0 1.0) (* (sin re) (fma 0.5 (* im im) 1.0)) (* (cosh im) 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 = cosh(im) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else if (t_0 <= 1.0) {
		tmp = sin(re) * fma(0.5, (im * im), 1.0);
	} else {
		tmp = cosh(im) * 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(cosh(im) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(re) * fma(0.5, Float64(im * im), 1.0));
	else
		tmp = Float64(cosh(im) * 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[Cosh[im], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $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[(N[Cosh[im], $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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 73.0

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

   7. Simplified 73.0%

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

   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. *-lowering-*.f64 N/A

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

      8. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 98.7

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

   5. Simplified 98.7%

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

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   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

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

   7. Simplified 80.6%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{re} \]
   8. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 80.6

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

   9. Applied egg-rr 80.6%

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

4. Final simplification 86.9%

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

Alternative 4: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma
        re
        (* re (fma (* re re) -0.0001984126984126984 0.008333333333333333))
        -0.16666666666666666)
       (* re (* re re))
       re)
      (fma
       (fma
        (* im im)
        (fma im (* im 0.001388888888888889) 0.041666666666666664)
        0.5)
       (* im im)
       1.0))
     (if (<= t_0 1.0) (* (sin re) (fma 0.5 (* im im) 1.0)) (* (cosh im) 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(fma(re, (re * fma((re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (re * (re * re)), re) * fma(fma((im * im), fma(im, (im * 0.001388888888888889), 0.041666666666666664), 0.5), (im * im), 1.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(re) * fma(0.5, (im * im), 1.0);
	} else {
		tmp = cosh(im) * 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(fma(re, Float64(re * fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(re * Float64(re * re)), re) * fma(fma(Float64(im * im), fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), 0.5), Float64(im * im), 1.0));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(re) * fma(0.5, Float64(im * im), 1.0));
	else
		tmp = Float64(cosh(im) * 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[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $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[(N[Cosh[im], $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 88.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      5. pow-plus N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{re}^{\left(2 + 1\right)}} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot {re}^{\color{blue}{3}} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. cube-unmult N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left(re \cdot {re}^{2}\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, re \cdot {re}^{2}, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

   8. Simplified 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right) \cdot \frac{1}{720}} + \frac{1}{24}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(\left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)} + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right)} \cdot \left(im \cdot im\right) + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\left(\left(\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) \cdot \left(im \cdot im\right) + \frac{1}{2} \cdot \left(im \cdot im\right)\right) + 1\right)} \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)} + 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right) \cdot \left(im \cdot im\right)} + 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}, im \cdot im, 1\right)} \]

   10. Applied egg-rr 63.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), im \cdot im, 1\right)} \]

   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. *-lowering-*.f64 N/A

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

      8. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 98.7

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

   5. Simplified 98.7%

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

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   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

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

   7. Simplified 80.6%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{re} \]
   8. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 80.6

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

   9. Applied egg-rr 80.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma
        re
        (* re (fma (* re re) -0.0001984126984126984 0.008333333333333333))
        -0.16666666666666666)
       (* re (* re re))
       re)
      (fma
       (fma
        (* im im)
        (fma im (* im 0.001388888888888889) 0.041666666666666664)
        0.5)
       (* im im)
       1.0))
     (if (<= t_0 1.0) (sin re) (* (cosh im) 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(fma(re, (re * fma((re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (re * (re * re)), re) * fma(fma((im * im), fma(im, (im * 0.001388888888888889), 0.041666666666666664), 0.5), (im * im), 1.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = cosh(im) * 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(fma(re, Float64(re * fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(re * Float64(re * re)), re) * fma(fma(Float64(im * im), fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), 0.5), Float64(im * im), 1.0));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(cosh(im) * 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[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[Cosh[im], $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 88.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      5. pow-plus N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{re}^{\left(2 + 1\right)}} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot {re}^{\color{blue}{3}} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. cube-unmult N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left(re \cdot {re}^{2}\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, re \cdot {re}^{2}, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

   8. Simplified 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right) \cdot \frac{1}{720}} + \frac{1}{24}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(\left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)} + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right)} \cdot \left(im \cdot im\right) + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\left(\left(\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) \cdot \left(im \cdot im\right) + \frac{1}{2} \cdot \left(im \cdot im\right)\right) + 1\right)} \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)} + 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right) \cdot \left(im \cdot im\right)} + 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}, im \cdot im, 1\right)} \]

   10. Applied egg-rr 63.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), im \cdot im, 1\right)} \]

   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. sin-lowering-sin.f64 97.6

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

   5. Simplified 97.6%

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

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   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

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

   7. Simplified 80.6%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{re} \]
   8. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 80.6

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

   9. Applied egg-rr 80.6%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, t\_1, 0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(t\_1, im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))))
        (t_1 (fma im (* im 0.001388888888888889) 0.041666666666666664)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (fma
        re
        (* re (fma (* re re) -0.0001984126984126984 0.008333333333333333))
        -0.16666666666666666)
       (* re (* re re))
       re)
      (fma (fma (* im im) t_1 0.5) (* im im) 1.0))
     (if (<= t_0 1.0)
       (sin re)
       (*
        (fma
         (* re re)
         (* re (fma (* re re) 0.008333333333333333 -0.16666666666666666))
         re)
        (fma t_1 (* im (* im (* im im))) (fma 0.5 (* im im) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	double t_1 = fma(im, (im * 0.001388888888888889), 0.041666666666666664);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(re, (re * fma((re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (re * (re * re)), re) * fma(fma((im * im), t_1, 0.5), (im * im), 1.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = fma((re * re), (re * fma((re * re), 0.008333333333333333, -0.16666666666666666)), re) * fma(t_1, (im * (im * (im * im))), fma(0.5, (im * im), 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	t_1 = fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(re, Float64(re * fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(re * Float64(re * re)), re) * fma(fma(Float64(im * im), t_1, 0.5), Float64(im * im), 1.0));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(fma(Float64(re * re), Float64(re * fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666)), re) * fma(t_1, Float64(im * Float64(im * Float64(im * im))), fma(0.5, Float64(im * im), 1.0)));
	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]}, Block[{t$95$1 = N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * t$95$1 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(t$95$1 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 88.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      5. pow-plus N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{re}^{\left(2 + 1\right)}} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot {re}^{\color{blue}{3}} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. cube-unmult N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}\right) \cdot \left(re \cdot {re}^{2}\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {re}^{2}\right) - \frac{1}{6}, re \cdot {re}^{2}, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

   8. Simplified 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right) \cdot \frac{1}{720}} + \frac{1}{24}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)} + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(\left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)} + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right)} \cdot \left(im \cdot im\right) + \left(\frac{1}{2} \cdot \left(im \cdot im\right) + 1\right)\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\left(\left(\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) \cdot \left(im \cdot im\right) + \frac{1}{2} \cdot \left(im \cdot im\right)\right) + 1\right)} \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right)} + 1\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \left(\color{blue}{\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}\right) \cdot \left(im \cdot im\right)} + 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{720} + \frac{1}{24}\right) + \frac{1}{2}, im \cdot im, 1\right)} \]

   10. Applied egg-rr 63.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), im \cdot im, 1\right)} \]

   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. sin-lowering-sin.f64 97.6

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

   5. Simplified 97.6%

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

   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 87.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re\right)} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left({re}^{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)} \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      14. *-lowering-*.f64 72.9

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.008333333333333333, -0.16666666666666666\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   8. Simplified 72.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), im \cdot im, 1\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 1.0) {
		tmp = sin(re) * fma((im * im), fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = cosh(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))) <= 1.0)
		tmp = Float64(sin(re) * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(cosh(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], 1.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 95.1

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

   7. Simplified 95.1%

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

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   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

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

   7. Simplified 80.6%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{re} \]
   8. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. cosh-lowering-cosh.f64 80.6

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

   9. Applied egg-rr 80.6%

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

4. Final simplification 91.6%

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

Alternative 8: 46.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.002)
   (*
    (* re (* re re))
    (fma (* im im) -0.08333333333333333 -0.16666666666666666))
   (*
    re
    (fma
     im
     (*
      im
      (fma
       (* im im)
       (fma im (* im 0.001388888888888889) 0.041666666666666664)
       0.5))
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.002) {
		tmp = (re * (re * re)) * fma((im * im), -0.08333333333333333, -0.16666666666666666);
	} else {
		tmp = re * fma(im, (im * fma((im * im), fma(im, (im * 0.001388888888888889), 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.002)
		tmp = Float64(Float64(re * Float64(re * re)) * fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666));
	else
		tmp = Float64(re * fma(im, Float64(im * fma(Float64(im * im), fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), 0.5)), 1.0));
	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], -0.002], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $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] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   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. *-lowering-*.f64 N/A

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

      8. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.4

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

   5. Simplified 72.4%

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

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 33.4

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

   8. Simplified 33.4%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. distribute-rgt-in N/A

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

       11. metadata-eval N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 N/A

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

       18. metadata-eval 11.9

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, -0.16666666666666666\right) \]

   11. Simplified 11.9%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 94.3%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 71.7%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   10. Applied egg-rr 71.7%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.002)
   (*
    (* re (* re re))
    (fma (* im im) -0.08333333333333333 -0.16666666666666666))
   (*
    re
    (fma
     im
     (* im (fma im (* im (* (* im im) 0.001388888888888889)) 0.5))
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.002) {
		tmp = (re * (re * re)) * fma((im * im), -0.08333333333333333, -0.16666666666666666);
	} else {
		tmp = re * fma(im, (im * fma(im, (im * ((im * im) * 0.001388888888888889)), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.002)
		tmp = Float64(Float64(re * Float64(re * re)) * fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666));
	else
		tmp = Float64(re * fma(im, Float64(im * fma(im, Float64(im * Float64(Float64(im * im) * 0.001388888888888889)), 0.5)), 1.0));
	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], -0.002], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(re * N[(im * N[(im * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   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. *-lowering-*.f64 N/A

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

      8. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.4

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

   5. Simplified 72.4%

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

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 33.4

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

   8. Simplified 33.4%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. distribute-rgt-in N/A

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

       11. metadata-eval N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 N/A

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

       18. metadata-eval 11.9

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, -0.16666666666666666\right) \]

   11. Simplified 11.9%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 94.3%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 71.7%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 71.7

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

   11. Simplified 71.7%

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

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

   14. Simplified 71.7%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \left(\left(im \cdot im\right) \cdot 0.001388888888888889\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.002)
   (*
    (* re (* re re))
    (fma (* im im) -0.08333333333333333 -0.16666666666666666))
   (* re (fma im (* im (fma im (* im 0.041666666666666664) 0.5)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.002) {
		tmp = (re * (re * re)) * fma((im * im), -0.08333333333333333, -0.16666666666666666);
	} else {
		tmp = re * fma(im, (im * fma(im, (im * 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.002)
		tmp = Float64(Float64(re * Float64(re * re)) * fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666));
	else
		tmp = Float64(re * fma(im, Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5)), 1.0));
	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], -0.002], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(re * N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   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. *-lowering-*.f64 N/A

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

      8. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.4

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

   5. Simplified 72.4%

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

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 33.4

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

   8. Simplified 33.4%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. distribute-rgt-in N/A

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

       11. metadata-eval N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 N/A

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

       18. metadata-eval 11.9

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, -0.16666666666666666\right) \]

   11. Simplified 11.9%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 94.3%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 71.7%

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

   9. Taylor expanded in im around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   10. 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. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. accelerator-lowering-fma.f64 N/A

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

       11. *-lowering-*.f64 70.5

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

   11. Simplified 70.5%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \left(im \cdot im\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.002)
   (*
    (* re (* re re))
    (fma (* im im) -0.08333333333333333 -0.16666666666666666))
   (fma (* 0.5 (* im im)) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.002) {
		tmp = (re * (re * re)) * fma((im * im), -0.08333333333333333, -0.16666666666666666);
	} else {
		tmp = fma((0.5 * (im * im)), 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))) <= -0.002)
		tmp = Float64(Float64(re * Float64(re * re)) * fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666));
	else
		tmp = fma(Float64(0.5 * Float64(im * im)), 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], -0.002], N[(N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(im * im), $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))) < -2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   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. *-lowering-*.f64 N/A

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

      8. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 72.4

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

   5. Simplified 72.4%

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

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 33.4

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

   8. Simplified 33.4%

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

   9. Taylor expanded in re around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. distribute-rgt-in N/A

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

       11. metadata-eval N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. associate-*l* N/A

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

       15. accelerator-lowering-fma.f64 N/A

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 N/A

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

       18. metadata-eval 11.9

           \[\leadsto \left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.08333333333333333}, -0.16666666666666666\right) \]

   11. Simplified 11.9%

       \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   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. *-lowering-*.f64 N/A

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

      8. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
   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-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 63.6

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

   8. Simplified 63.6%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\left(re \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \left(im \cdot im\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \left(im \cdot im\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.002)
   (* re (* (* re re) -0.16666666666666666))
   (fma (* 0.5 (* im im)) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.002) {
		tmp = re * ((re * re) * -0.16666666666666666);
	} else {
		tmp = fma((0.5 * (im * im)), 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))) <= -0.002)
		tmp = Float64(re * Float64(Float64(re * re) * -0.16666666666666666));
	else
		tmp = fma(Float64(0.5 * Float64(im * im)), 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], -0.002], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(im * im), $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))) < -2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 30.4

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

   5. Simplified 30.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 8.8

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

   8. Simplified 8.8%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 8.5

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

   11. Simplified 8.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   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. *-lowering-*.f64 N/A

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

      8. sin-lowering-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. accelerator-lowering-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. *-lowering-*.f64 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
   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-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 63.6

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

   8. Simplified 63.6%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot \left(im \cdot im\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \left(re \cdot 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.002)
   (* re (* (* re re) -0.16666666666666666))
   (fma im (* im (* re 0.5)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.002) {
		tmp = re * ((re * re) * -0.16666666666666666);
	} else {
		tmp = fma(im, (im * (re * 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))) <= -0.002)
		tmp = Float64(re * Float64(Float64(re * re) * -0.16666666666666666));
	else
		tmp = fma(im, Float64(im * Float64(re * 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], -0.002], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(re * 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))) < -2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 30.4

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

   5. Simplified 30.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 8.8

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

   8. Simplified 8.8%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 8.5

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

   11. Simplified 8.5%

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

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. exp-0 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. exp-0 N/A

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

      13. cosh-lowering-cosh.f64 N/A

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

      14. sin-lowering-sin.f64 100.0

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

   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

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

   7. Simplified 75.5%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 56.9

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

   10. Simplified 56.9%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot \left(re \cdot 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.002)
   (* re (* (* re re) -0.16666666666666666))
   re))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.002) {
		tmp = re * ((re * re) * -0.16666666666666666);
	} else {
		tmp = re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((sin(re) * 0.5d0) * (exp(-im) + exp(im))) <= (-0.002d0)) then
        tmp = re * ((re * re) * (-0.16666666666666666d0))
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((Math.sin(re) * 0.5) * (Math.exp(-im) + Math.exp(im))) <= -0.002) {
		tmp = re * ((re * re) * -0.16666666666666666);
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((math.sin(re) * 0.5) * (math.exp(-im) + math.exp(im))) <= -0.002:
		tmp = re * ((re * re) * -0.16666666666666666)
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.002)
		tmp = Float64(re * Float64(Float64(re * re) * -0.16666666666666666));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.002)
		tmp = re * ((re * re) * -0.16666666666666666);
	else
		tmp = re;
	end
	tmp_2 = 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], -0.002], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], re]

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))) < -2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 30.4

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

   5. Simplified 30.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 8.8

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

   8. Simplified 8.8%

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

   9. Taylor expanded in re around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 8.5

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

   11. Simplified 8.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 58.7

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

   5. Simplified 58.7%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 42.9%

      \[\leadsto \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.002)
   (*
    (fma re (* (* re re) -0.16666666666666666) re)
    (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0))
   (*
    (fma
     (* re re)
     (* re (fma (* re re) 0.008333333333333333 -0.16666666666666666))
     re)
    (fma
     (fma im (* im 0.001388888888888889) 0.041666666666666664)
     (* im (* im (* im im)))
     (fma 0.5 (* im im) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.002) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	} else {
		tmp = fma((re * re), (re * fma((re * re), 0.008333333333333333, -0.16666666666666666)), re) * fma(fma(im, (im * 0.001388888888888889), 0.041666666666666664), (im * (im * (im * im))), fma(0.5, (im * im), 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.002)
		tmp = Float64(fma(re, Float64(Float64(re * re) * -0.16666666666666666), re) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = Float64(fma(Float64(re * re), Float64(re * fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666)), re) * fma(fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), Float64(im * Float64(im * Float64(im * im))), fma(0.5, Float64(im * im), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.002], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   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. *-lowering-*.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. sin-lowering-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 92.6%

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

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 22.2

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

   8. Simplified 22.2%

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

   if -2e-3 < (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 92.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re\right)} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      4. *-lft-identity N/A

         \[\leadsto \left({re}^{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)} \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{720}, \frac{1}{24}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(\frac{1}{2}, im \cdot im, 1\right)\right) \]

      14. *-lowering-*.f64 74.1

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.008333333333333333, -0.16666666666666666\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]

   8. Simplified 74.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.002)
   (*
    (fma re (* (* re re) -0.16666666666666666) re)
    (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0))
   (*
    re
    (fma
     im
     (*
      im
      (fma
       (* im im)
       (fma im (* im 0.001388888888888889) 0.041666666666666664)
       0.5))
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.002) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	} else {
		tmp = re * fma(im, (im * fma((im * im), fma(im, (im * 0.001388888888888889), 0.041666666666666664), 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.002)
		tmp = Float64(fma(re, Float64(Float64(re * re) * -0.16666666666666666), re) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = Float64(re * fma(im, Float64(im * fma(Float64(im * im), fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), 0.5)), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.002], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $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] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   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. *-lowering-*.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. sin-lowering-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 92.6%

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

   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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 22.2

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

   8. Simplified 22.2%

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

   if -2e-3 < (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 92.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 74.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

   10. Applied egg-rr 74.2%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.5% accurate, 317.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. sin-lowering-sin.f64 47.2

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

5. Simplified 47.2%

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

6. Taylor expanded in re around 0

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

8. Simplified 26.6%

   \[\leadsto \color{blue}{re} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024201 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))