math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.8s

Alternatives: 21

Speedup: 1.5×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-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. Step-by-step derivation

   1. *-commutative N/A

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

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

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

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

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

   4. *-lft-identity N/A

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

   5. cosh-lowering-cosh.f64 100.0

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

6. Applied egg-rr 100.0%

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

Alternative 2: 83.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   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-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 78.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 63.8

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

   8. Simplified 63.8%

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

   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

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   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-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 99.3%

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

   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. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
   8. Step-by-step derivation

   9. Simplified 81.7%

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

4. Final simplification 85.7%

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

Alternative 3: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot 0.5\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) 0.5)) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma re (* (* re re) -0.16666666666666666) re)
      (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
     (if (<= t_1 1.0) (* t_0 (fma im im 2.0)) (* re (cosh im))))))

C

double code(double re, double im) {
	double t_0 = sin(re) * 0.5;
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else if (t_1 <= 1.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = re * cosh(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * 0.5)
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(Float64(re * re) * -0.16666666666666666), re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(re * cosh(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   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-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 78.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 63.8

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

   8. Simplified 63.8%

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

   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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 99.3

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

   5. Simplified 99.3%

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

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

   1. Initial program 100.0%

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

      1. *-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. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
   8. Step-by-step derivation

   9. Simplified 81.7%

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

4. Final simplification 85.7%

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

Alternative 4: 83.5% accurate, 0.4× speedup

Math

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

FPCore

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

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(re, ((re * re) * -0.16666666666666666), re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = re * cosh(im);
	}
	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(re, Float64(Float64(re * re) * -0.16666666666666666), re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(re * cosh(im));
	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[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   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-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 78.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 63.8

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

   8. Simplified 63.8%

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

   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 98.7

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

   5. Simplified 98.7%

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

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
   8. Step-by-step derivation

   9. Simplified 81.7%

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

4. Final simplification 85.4%

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

Alternative 5: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right)\right), 2\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (* (* re re) -0.16666666666666666) re)
      (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
     (if (<= t_0 1.0)
       (sin re)
       (*
        (fma
         (* im im)
         (* im (* im (fma (* im im) 0.002777777777777778 0.08333333333333333)))
         2.0)
        (* re 0.5))))))

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(re, ((re * re) * -0.16666666666666666), re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = fma((im * im), (im * (im * fma((im * im), 0.002777777777777778, 0.08333333333333333))), 2.0) * (re * 0.5);
	}
	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(re, Float64(Float64(re * re) * -0.16666666666666666), re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(fma(Float64(im * im), Float64(im * Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333))), 2.0) * Float64(re * 0.5));
	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[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(re * 0.5), $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

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   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-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 78.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 63.8

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

   8. Simplified 63.8%

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

   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 98.7

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

   5. Simplified 98.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   5. Simplified 82.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(re \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re\right)} \]

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

         \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re\right)} \]

   8. Simplified 66.7%

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

   9. Taylor expanded in im around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

       6. associate-*l/ N/A

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

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. associate-/l* N/A

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

       10. *-rgt-identity N/A

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

       11. associate-*r/ N/A

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

       12. rgt-mult-inverse N/A

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

       13. *-rgt-identity N/A

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

       14. *-commutative N/A

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

       15. *-commutative N/A

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

       16. metadata-eval N/A

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

       17. pow-sqr N/A

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

       18. associate-*l* N/A

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

       19. distribute-rgt-in N/A

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

   11. Simplified 66.7%

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

4. Final simplification 81.2%

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

Alternative 6: 52.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot 0.5, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.05)
     (* re (* (* im im) (fma -0.08333333333333333 (* re re) 0.5)))
     (if (<= t_0 1.0)
       (fma (* im im) (* re 0.5) re)
       (* re (* im (* im (* (* im im) 0.041666666666666664))))))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.05) {
		tmp = re * ((im * im) * fma(-0.08333333333333333, (re * re), 0.5));
	} else if (t_0 <= 1.0) {
		tmp = fma((im * im), (re * 0.5), re);
	} else {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	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 <= -0.05)
		tmp = Float64(re * Float64(Float64(im * im) * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	elseif (t_0 <= 1.0)
		tmp = fma(Float64(im * im), Float64(re * 0.5), re);
	else
		tmp = Float64(re * Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664))));
	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, -0.05], N[(re * N[(N[(im * im), $MachinePrecision] * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(im * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $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))) < -0.050000000000000003

   1. Initial program 100.0%

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

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.3

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

   7. Simplified 69.3%

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

   8. Taylor expanded in im around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 35.4

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

   10. Simplified 35.4%

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

   11. Taylor expanded in re around 0

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

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

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 31.7

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

   13. Simplified 31.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(re \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re\right)} \]

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

         \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re\right)} \]

   8. Simplified 61.9%

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

   9. Taylor expanded in im around 0

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

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

       6. unpow2 N/A

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

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

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

       8. *-lowering-*.f64 61.9

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

   11. Simplified 61.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   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-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 77.0%

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

   6. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 53.4

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

   8. Simplified 53.4%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

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

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

       12. *-commutative N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 64.1

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

   11. Simplified 64.1%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot 0.5, re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.05)
     (* re (* (* re re) -0.16666666666666666))
     (if (<= t_0 1.0) re (* re (* 0.5 (* im im)))))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.05) {
		tmp = re * ((re * re) * -0.16666666666666666);
	} else if (t_0 <= 1.0) {
		tmp = re;
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(re) * 0.5d0) * (exp(-im) + exp(im))
    if (t_0 <= (-0.05d0)) then
        tmp = re * ((re * re) * (-0.16666666666666666d0))
    else if (t_0 <= 1.0d0) then
        tmp = re
    else
        tmp = re * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.sin(re) * 0.5) * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_0 <= -0.05) {
		tmp = re * ((re * re) * -0.16666666666666666);
	} else if (t_0 <= 1.0) {
		tmp = re;
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.sin(re) * 0.5) * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_0 <= -0.05:
		tmp = re * ((re * re) * -0.16666666666666666)
	elif t_0 <= 1.0:
		tmp = re
	else:
		tmp = re * (0.5 * (im * im))
	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 <= -0.05)
		tmp = Float64(re * Float64(Float64(re * re) * -0.16666666666666666));
	elseif (t_0 <= 1.0)
		tmp = re;
	else
		tmp = Float64(re * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_0 <= -0.05)
		tmp = re * ((re * re) * -0.16666666666666666);
	elseif (t_0 <= 1.0)
		tmp = re;
	else
		tmp = re * (0.5 * (im * im));
	end
	tmp_2 = 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, -0.05], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], re, N[(re * N[(0.5 * N[(im * im), $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))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 36.8

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

   5. Simplified 36.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + 1 \cdot re \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + 1 \cdot re \]

      5. pow-plus N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{\left(2 + 1\right)}} + 1 \cdot re \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{\color{blue}{3}} + 1 \cdot re \]

      7. cube-unmult N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} + 1 \cdot re \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) + 1 \cdot re \]

      9. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(re \cdot {re}^{2}\right) + \color{blue}{re} \]

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

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

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

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

      15. unpow2 N/A

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

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

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 16.7

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

   8. Simplified 16.7%

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

   9. Taylor expanded in re around 0

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

   11. Simplified 12.9%

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

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   13. 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 12.6

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

   14. Simplified 12.6%

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

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

   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 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 61.5%

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

   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 im around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 52.5

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

   7. Simplified 52.5%

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

   8. Taylor expanded in im around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 52.5

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

   10. Simplified 52.5%

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

   11. Taylor expanded in re around 0

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

   13. Simplified 43.6%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.7% accurate, 0.7× speedup

Math

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

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))
   (* re (cosh im))))

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 = re * cosh(im);
	}
	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(re * cosh(im));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 94.4

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

   9. Simplified 94.4%

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

   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. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
   8. Step-by-step derivation

   9. Simplified 81.7%

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

4. Final simplification 90.9%

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

Alternative 9: 89.6% accurate, 0.7× speedup

Math

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

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) (* (* im im) 0.001388888888888889) 0.5) 1.0))
   (* re (cosh im))))

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), ((im * im) * 0.001388888888888889), 0.5), 1.0);
	} else {
		tmp = re * cosh(im);
	}
	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), Float64(Float64(im * im) * 0.001388888888888889), 0.5), 1.0));
	else
		tmp = Float64(re * cosh(im));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 94.4

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

   9. Simplified 94.4%

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

   10. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 94.4

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

   12. Simplified 94.4%

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

   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. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
   8. Step-by-step derivation

   9. Simplified 81.7%

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

4. Final simplification 90.9%

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

Alternative 10: 89.3% accurate, 0.7× speedup

Math

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

FPCore

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

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), (im * (im * ((im * im) * 0.001388888888888889))), 1.0);
	} else {
		tmp = re * cosh(im);
	}
	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), Float64(im * Float64(im * Float64(Float64(im * im) * 0.001388888888888889))), 1.0));
	else
		tmp = Float64(re * cosh(im));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 94.4

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

   9. Simplified 94.4%

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

   10. Taylor expanded in im around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 94.0

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

   12. Simplified 94.0%

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

   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. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
   8. Step-by-step derivation

   9. Simplified 81.7%

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

4. Final simplification 90.6%

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

Alternative 11: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right)\right), 2\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 5e-7)
   (*
    (fma re (* (* re re) -0.16666666666666666) re)
    (fma (* im im) (fma im (* im 0.041666666666666664) 0.5) 1.0))
   (*
    (fma
     (* im im)
     (* im (* im (fma (* im im) 0.002777777777777778 0.08333333333333333)))
     2.0)
    (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 5e-7) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re) * fma((im * im), fma(im, (im * 0.041666666666666664), 0.5), 1.0);
	} else {
		tmp = fma((im * im), (im * (im * fma((im * im), 0.002777777777777778, 0.08333333333333333))), 2.0) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 5e-7)
		tmp = Float64(fma(re, Float64(Float64(re * re) * -0.16666666666666666), re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(fma(Float64(im * im), Float64(im * Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333))), 2.0) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.99999999999999977e-7

   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-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 90.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 61.6

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

   8. Simplified 61.6%

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

   if 4.99999999999999977e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   5. Simplified 88.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(re \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re\right)} \]

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

         \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re\right)} \]

   8. Simplified 46.4%

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

   9. Taylor expanded in im around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

       6. associate-*l/ N/A

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

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. associate-/l* N/A

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

       10. *-rgt-identity N/A

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

       11. associate-*r/ N/A

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

       12. rgt-mult-inverse N/A

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

       13. *-rgt-identity N/A

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

       14. *-commutative N/A

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

       15. *-commutative N/A

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

       16. metadata-eval N/A

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

       17. pow-sqr N/A

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

       18. associate-*l* N/A

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

       19. distribute-rgt-in N/A

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

   11. Simplified 46.4%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right)\right), 2\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right)\right), 2\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 5e-7)
   (* re (* (fma re (* re -0.16666666666666666) 1.0) (fma 0.5 (* im im) 1.0)))
   (*
    (fma
     (* im im)
     (* im (* im (fma (* im im) 0.002777777777777778 0.08333333333333333)))
     2.0)
    (* re 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 5e-7) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma(0.5, (im * im), 1.0));
	} else {
		tmp = fma((im * im), (im * (im * fma((im * im), 0.002777777777777778, 0.08333333333333333))), 2.0) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 5e-7)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(0.5, Float64(im * im), 1.0)));
	else
		tmp = Float64(fma(Float64(im * im), Float64(im * Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333))), 2.0) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-7], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.99999999999999977e-7

   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

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 79.8

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

   7. Simplified 79.8%

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

   8. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 55.2

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

   10. Simplified 55.2%

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

   if 4.99999999999999977e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

   5. Simplified 88.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(re \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re\right)} \]

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

         \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re\right)} \]

   8. Simplified 46.4%

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

   9. Taylor expanded in im around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

       6. associate-*l/ N/A

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

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. associate-/l* N/A

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

       10. *-rgt-identity N/A

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

       11. associate-*r/ N/A

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

       12. rgt-mult-inverse N/A

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

       13. *-rgt-identity N/A

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

       14. *-commutative N/A

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

       15. *-commutative N/A

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

       16. metadata-eval N/A

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

       17. pow-sqr N/A

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

       18. associate-*l* N/A

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

       19. distribute-rgt-in N/A

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

   11. Simplified 46.4%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right)\right), 2\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 5e-7)
   (* re (* (fma re (* re -0.16666666666666666) 1.0) (fma 0.5 (* im im) 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) * 0.5) * (exp(-im) + exp(im))) <= 5e-7) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma(0.5, (im * im), 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 (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 5e-7)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(0.5, Float64(im * im), 1.0)));
	else
		tmp = Float64(re * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * 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], 5e-7], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $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))) < 4.99999999999999977e-7

   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

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 79.8

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

   7. Simplified 79.8%

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

   8. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 55.2

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

   10. Simplified 55.2%

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

   if 4.99999999999999977e-7 < (*.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. Step-by-step derivation

      1. *-commutative N/A

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

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

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

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

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

      4. *-lft-identity N/A

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

      5. cosh-lowering-cosh.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 88.1

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

   9. Simplified 88.1%

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

   10. Taylor expanded in re around 0

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

   12. Simplified 46.4%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 5e-7)
   (* re (* (fma re (* re -0.16666666666666666) 1.0) (fma 0.5 (* im im) 1.0)))
   (* 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))) <= 5e-7) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma(0.5, (im * im), 1.0));
	} 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))) <= 5e-7)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(0.5, Float64(im * im), 1.0)));
	else
		tmp = Float64(re * fma(Float64(im * 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], 5e-7], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $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))) < 4.99999999999999977e-7

   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

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 79.8

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

   7. Simplified 79.8%

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

   8. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 55.2

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

   10. Simplified 55.2%

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

   if 4.99999999999999977e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 84.5%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 44.6%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.05)
   (* re (* (* im im) (fma -0.08333333333333333 (* re re) 0.5)))
   (* 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))) <= -0.05) {
		tmp = re * ((im * im) * fma(-0.08333333333333333, (re * re), 0.5));
	} 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))) <= -0.05)
		tmp = Float64(re * Float64(Float64(im * im) * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	else
		tmp = Float64(re * fma(Float64(im * 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], -0.05], N[(re * N[(N[(im * im), $MachinePrecision] * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $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))) < -0.050000000000000003

   1. Initial program 100.0%

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

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.3

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

   7. Simplified 69.3%

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

   8. Taylor expanded in im around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 35.4

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

   10. Simplified 35.4%

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

   11. Taylor expanded in re around 0

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

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

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 31.7

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

   13. Simplified 31.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   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-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft1-in N/A

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

   5. Simplified 89.6%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 62.9%

      \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.54:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 0.54)
   (fma -0.16666666666666666 (* re (* re re)) re)
   (* re (* im (* im (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 0.54) {
		tmp = fma(-0.16666666666666666, (re * (re * re)), re);
	} else {
		tmp = re * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 0.54)
		tmp = fma(-0.16666666666666666, Float64(re * Float64(re * re)), re);
	else
		tmp = Float64(re * Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664))));
	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], 0.54], N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.54000000000000004

   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 62.3

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 62.3%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + 1 \cdot re \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + 1 \cdot re \]

      5. pow-plus N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{\left(2 + 1\right)}} + 1 \cdot re \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{\color{blue}{3}} + 1 \cdot re \]

      7. cube-unmult N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} + 1 \cdot re \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) + 1 \cdot re \]

      9. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(re \cdot {re}^{2}\right) + \color{blue}{re} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re \cdot {re}^{2}, re\right)} \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, re \cdot {re}^{2}, re\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), re \cdot {re}^{2}, re\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, re \cdot {re}^{2}, re\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, re \cdot {re}^{2}, re\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), re \cdot {re}^{2}, re\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), re \cdot {re}^{2}, re\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot {re}^{2}}, re\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

      19. *-lowering-*.f64 41.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

   8. Simplified 41.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{6}}, re \cdot \left(re \cdot re\right), re\right) \]
   10. Step-by-step derivation

   11. Simplified 39.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{-0.16666666666666666}, re \cdot \left(re \cdot re\right), re\right) \]

   if 0.54000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{24}\right)} \cdot {im}^{2} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re} + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right) \]

      9. associate-*r* N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\frac{1}{2} \cdot \left(\sin re \cdot {im}^{2}\right)} + \sin re\right) \]

      10. *-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\frac{1}{2} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} + \sin re\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \left(\color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} + \sin re\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right), 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{1 \cdot re + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{re} + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re + re} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot re\right)} + re \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot re, re\right)} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot re, re\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot re, re\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot re}, re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot re, re\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \left(\color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}\right) \cdot re, re\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)} \cdot re, re\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right) \cdot re, re\right) \]

      13. *-lowering-*.f64 43.3

          \[\leadsto \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right) \cdot re, re\right) \]

   8. Simplified 43.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot re, re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot re\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right) \cdot re} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{24} \cdot {im}^{4}\right)} \]

       3. metadata-eval N/A

          \[\leadsto re \cdot \left(\frac{1}{24} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \]

       4. pow-sqr N/A

          \[\leadsto re \cdot \left(\frac{1}{24} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       5. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}\right)} \]

       6. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{re \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

       8. unpow2 N/A

          \[\leadsto re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right)} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto re \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

       12. *-commutative N/A

           \[\leadsto re \cdot \left(im \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto re \cdot \left(im \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto re \cdot \left(im \cdot \left(im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{24}\right)\right)\right) \]

       15. *-lowering-*.f64 51.8

           \[\leadsto re \cdot \left(im \cdot \left(im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.041666666666666664\right)\right)\right) \]

   11. Simplified 51.8%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.54:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 41.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 5e-7)
   (fma -0.16666666666666666 (* re (* re re)) re)
   (* re (* 0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 5e-7) {
		tmp = fma(-0.16666666666666666, (re * (re * re)), re);
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 5e-7)
		tmp = fma(-0.16666666666666666, Float64(re * Float64(re * re)), re);
	else
		tmp = Float64(re * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-7], N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.99999999999999977e-7

   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 58.3

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 58.3%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + 1 \cdot re \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + 1 \cdot re \]

      5. pow-plus N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{\left(2 + 1\right)}} + 1 \cdot re \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{\color{blue}{3}} + 1 \cdot re \]

      7. cube-unmult N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} + 1 \cdot re \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) + 1 \cdot re \]

      9. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(re \cdot {re}^{2}\right) + \color{blue}{re} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re \cdot {re}^{2}, re\right)} \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, re \cdot {re}^{2}, re\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), re \cdot {re}^{2}, re\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, re \cdot {re}^{2}, re\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, re \cdot {re}^{2}, re\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), re \cdot {re}^{2}, re\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), re \cdot {re}^{2}, re\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot {re}^{2}}, re\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

      19. *-lowering-*.f64 45.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

   8. Simplified 45.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{6}}, re \cdot \left(re \cdot re\right), re\right) \]
   10. Step-by-step derivation

   11. Simplified 42.7%

       \[\leadsto \mathsf{fma}\left(\color{blue}{-0.16666666666666666}, re \cdot \left(re \cdot re\right), re\right) \]

   if 4.99999999999999977e-7 < (*.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 im around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \cdot \sin re \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {im}^{2}, 1\right)} \cdot \sin re \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      4. *-lowering-*.f64 67.9

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Simplified 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right)} \cdot \sin re \]

   8. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)} \cdot \sin re \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)} \cdot \sin re \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

      3. *-lowering-*.f64 36.7

         \[\leadsto \left(0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \sin re \]

   10. Simplified 36.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot \left(im \cdot im\right)\right)} \cdot \sin re \]

   11. Taylor expanded in re around 0

       \[\leadsto \left(\frac{1}{2} \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{re} \]
   12. Step-by-step derivation

   13. Simplified 30.7%

       \[\leadsto \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 30.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.05)
   (* re (* (* re re) -0.16666666666666666))
   re))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.05) {
		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))) <= (-0.05d0)) 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))) <= -0.05) {
		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))) <= -0.05:
		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))) <= -0.05)
		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))) <= -0.05)
		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], -0.05], 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))) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 36.8

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 36.8%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + 1 \cdot re \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + 1 \cdot re \]

      5. pow-plus N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{\left(2 + 1\right)}} + 1 \cdot re \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{\color{blue}{3}} + 1 \cdot re \]

      7. cube-unmult N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} + 1 \cdot re \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) + 1 \cdot re \]

      9. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(re \cdot {re}^{2}\right) + \color{blue}{re} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re \cdot {re}^{2}, re\right)} \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, re \cdot {re}^{2}, re\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), re \cdot {re}^{2}, re\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, re \cdot {re}^{2}, re\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, re \cdot {re}^{2}, re\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), re \cdot {re}^{2}, re\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), re \cdot {re}^{2}, re\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot {re}^{2}}, re\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

      19. *-lowering-*.f64 16.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

   8. Simplified 16.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{6}}, re \cdot \left(re \cdot re\right), re\right) \]
   10. Step-by-step derivation

   11. Simplified 12.9%

       \[\leadsto \mathsf{fma}\left(\color{blue}{-0.16666666666666666}, re \cdot \left(re \cdot re\right), re\right) \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   13. 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 12.6

          \[\leadsto re \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   14. Simplified 12.6%

       \[\leadsto \color{blue}{re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 55.7

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 55.7%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \]
   7. Step-by-step derivation

   8. Simplified 34.7%

      \[\leadsto \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), 1\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 5e-7)
   (*
    (fma
     (* im im)
     (fma
      (* im im)
      (fma (* im im) 0.002777777777777778 0.08333333333333333)
      1.0)
     2.0)
    (* re (fma re (* re -0.08333333333333333) 0.5)))
   (*
    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) <= 5e-7) {
		tmp = fma((im * im), fma((im * im), fma((im * im), 0.002777777777777778, 0.08333333333333333), 1.0), 2.0) * (re * fma(re, (re * -0.08333333333333333), 0.5));
	} 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) <= 5e-7)
		tmp = Float64(fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333), 1.0), 2.0) * Float64(re * fma(re, Float64(re * -0.08333333333333333), 0.5)));
	else
		tmp = Float64(re * fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 5e-7], N[(N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision] * N[(re * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * 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]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 4.99999999999999977e-7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Simplified 89.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) + \frac{1}{2} \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), 1\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)} \]

   if 4.99999999999999977e-7 < (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. 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. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(1 \cdot \cosh im\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(1 \cdot \cosh im\right)} \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(1 \cdot \cosh im\right) \]

      4. *-lft-identity N/A

         \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

      5. cosh-lowering-cosh.f64 100.0

         \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   7. Taylor expanded in im around 0

      \[\leadsto \sin re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), 1\right) \]

      5. +-commutative N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, 1\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, \frac{1}{2}\right), 1\right) \]

      9. +-commutative N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \]

      10. *-commutative N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \]

      12. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]

      13. *-lowering-*.f64 95.0

          \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \]

   9. Simplified 95.0%

      \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

   10. Taylor expanded in re around 0

       \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \]
   11. Step-by-step derivation

   12. Simplified 18.2%

       \[\leadsto \color{blue}{re} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 47.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot 0.5, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.05)
   (* re (* (* re re) -0.16666666666666666))
   (fma (* im im) (* re 0.5) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.05) {
		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 (sin(re) <= -0.05)
		tmp = Float64(re * Float64(Float64(re * re) * -0.16666666666666666));
	else
		tmp = fma(Float64(im * im), Float64(re * 0.5), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.05], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * 0.5), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 51.1

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 51.1%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + 1 \cdot re \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + 1 \cdot re \]

      5. pow-plus N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{\left(2 + 1\right)}} + 1 \cdot re \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{\color{blue}{3}} + 1 \cdot re \]

      7. cube-unmult N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} + 1 \cdot re \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(re \cdot \color{blue}{{re}^{2}}\right) + 1 \cdot re \]

      9. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(re \cdot {re}^{2}\right) + \color{blue}{re} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re \cdot {re}^{2}, re\right)} \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, re \cdot {re}^{2}, re\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), re \cdot {re}^{2}, re\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, re \cdot {re}^{2}, re\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, re \cdot {re}^{2}, re\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), re \cdot {re}^{2}, re\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), re \cdot {re}^{2}, re\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot {re}^{2}}, re\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

      19. *-lowering-*.f64 22.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \]

   8. Simplified 22.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{6}}, re \cdot \left(re \cdot re\right), re\right) \]
   10. Step-by-step derivation

   11. Simplified 17.3%

       \[\leadsto \mathsf{fma}\left(\color{blue}{-0.16666666666666666}, re \cdot \left(re \cdot re\right), re\right) \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   13. 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 17.3

          \[\leadsto re \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   14. Simplified 17.3%

       \[\leadsto \color{blue}{re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)} \]

   if -0.050000000000000003 < (sin.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right), 2\right)} \]

   5. Simplified 91.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(re \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right) \cdot \left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 + im \cdot \left(im + {im}^{3} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot re\right)} \]

   8. Simplified 67.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), 1\right), 2\right) \cdot \left(0.5 \cdot re\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{re + \frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} \]
   10. 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. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{2} \cdot re, re\right)} \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} \cdot re, re\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{2} \cdot re, re\right) \]

       8. *-lowering-*.f64 53.6

          \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{0.5 \cdot re}, re\right) \]

   11. Simplified 53.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, 0.5 \cdot re, re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot 0.5, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 27.5% 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 48.4

      \[\leadsto \color{blue}{\sin re} \]

5. Simplified 48.4%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re} \]
7. Step-by-step derivation

8. Simplified 22.4%

   \[\leadsto \color{blue}{re} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))