math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.9s

Alternatives: 22

Speedup: 1.5×

• 50.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right)\\ t_1 := \mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_1, im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \left(\left(im \cdot im\right) \cdot im\right) \cdot im, \mathsf{fma}\left(im \cdot 0.5, im, 1\right)\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, -1\right), im, 1\right) + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp im) (exp (- im))) (* (sin re) 0.5)))
        (t_1 (fma 0.001388888888888889 (* im im) 0.041666666666666664)))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma (* (fma t_1 (* im im) 0.5) im) im 1.0)
       (* (* re re) -0.16666666666666666))
      re)
     (if (<= t_0 5e+36)
       (* (fma t_1 (* (* (* im im) im) im) (fma (* im 0.5) im 1.0)) (sin re))
       (* (+ (fma (fma im 0.5 -1.0) im 1.0) (exp im)) (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = (exp(im) + exp(-im)) * (sin(re) * 0.5);
	double t_1 = fma(0.001388888888888889, (im * im), 0.041666666666666664);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma((fma(t_1, (im * im), 0.5) * im), im, 1.0) * ((re * re) * -0.16666666666666666)) * re;
	} else if (t_0 <= 5e+36) {
		tmp = fma(t_1, (((im * im) * im) * im), fma((im * 0.5), im, 1.0)) * sin(re);
	} else {
		tmp = (fma(fma(im, 0.5, -1.0), im, 1.0) + exp(im)) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(sin(re) * 0.5))
	t_1 = fma(0.001388888888888889, Float64(im * im), 0.041666666666666664)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(fma(t_1, Float64(im * im), 0.5) * im), im, 1.0) * Float64(Float64(re * re) * -0.16666666666666666)) * re);
	elseif (t_0 <= 5e+36)
		tmp = Float64(fma(t_1, Float64(Float64(Float64(im * im) * im) * im), fma(Float64(im * 0.5), im, 1.0)) * sin(re));
	else
		tmp = Float64(Float64(fma(fma(im, 0.5, -1.0), im, 1.0) + exp(im)) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(t$95$1 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 5e+36], N[(N[(t$95$1 * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] + N[(N[(im * 0.5), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im * 0.5 + -1.0), $MachinePrecision] * im + 1.0), $MachinePrecision] + N[Exp[im], $MachinePrecision]), $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

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 35.6%

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

   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))) < 4.99999999999999977e36

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 98.8%

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

   if 4.99999999999999977e36 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.3

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

   8. Applied rewrites 65.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), \left(\left(im \cdot im\right) \cdot im\right) \cdot im, \mathsf{fma}\left(im \cdot 0.5, im, 1\right)\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, -1\right), im, 1\right) + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right)\\ t_1 := \left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+36}:\\ \;\;\;\;t\_0 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, -1\right), im, 1\right) + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (*
           (fma
            (fma 0.001388888888888889 (* im im) 0.041666666666666664)
            (* im im)
            0.5)
           im)
          im
          1.0))
        (t_1 (* (+ (exp im) (exp (- im))) (* (sin re) 0.5))))
   (if (<= t_1 (- INFINITY))
     (* (* t_0 (* (* re re) -0.16666666666666666)) re)
     (if (<= t_1 5e+36)
       (* t_0 (sin re))
       (* (+ (fma (fma im 0.5 -1.0) im 1.0) (exp im)) (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = fma((fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5) * im), im, 1.0);
	double t_1 = (exp(im) + exp(-im)) * (sin(re) * 0.5);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_0 * ((re * re) * -0.16666666666666666)) * re;
	} else if (t_1 <= 5e+36) {
		tmp = t_0 * sin(re);
	} else {
		tmp = (fma(fma(im, 0.5, -1.0), im, 1.0) + exp(im)) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5) * im), im, 1.0)
	t_1 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(Float64(re * re) * -0.16666666666666666)) * re);
	elseif (t_1 <= 5e+36)
		tmp = Float64(t_0 * sin(re));
	else
		tmp = Float64(Float64(fma(fma(im, 0.5, -1.0), im, 1.0) + exp(im)) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$1, 5e+36], N[(t$95$0 * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im * 0.5 + -1.0), $MachinePrecision] * im + 1.0), $MachinePrecision] + N[Exp[im], $MachinePrecision]), $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

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 35.6%

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

   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))) < 4.99999999999999977e36

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 98.8%

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

   if 4.99999999999999977e36 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.3

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

   8. Applied rewrites 65.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, -1\right), im, 1\right) + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, -1\right), im, 1\right) + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp im) (exp (- im))) (* (sin re) 0.5))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (*
         (fma
          (fma 0.001388888888888889 (* im im) 0.041666666666666664)
          (* im im)
          0.5)
         im)
        im
        1.0)
       (* (* re re) -0.16666666666666666))
      re)
     (if (<= t_0 5e+36)
       (*
        (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0)
        (sin re))
       (* (+ (fma (fma im 0.5 -1.0) im 1.0) (exp im)) (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = (exp(im) + exp(-im)) * (sin(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma((fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5) * im), im, 1.0) * ((re * re) * -0.16666666666666666)) * re;
	} else if (t_0 <= 5e+36) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else {
		tmp = (fma(fma(im, 0.5, -1.0), im, 1.0) + exp(im)) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5) * im), im, 1.0) * Float64(Float64(re * re) * -0.16666666666666666)) * re);
	elseif (t_0 <= 5e+36)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	else
		tmp = Float64(Float64(fma(fma(im, 0.5, -1.0), im, 1.0) + exp(im)) * Float64(re * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 5e+36], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im * 0.5 + -1.0), $MachinePrecision] * im + 1.0), $MachinePrecision] + N[Exp[im], $MachinePrecision]), $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

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 35.6%

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

   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))) < 4.99999999999999977e36

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*l* N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 4.99999999999999977e36 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.3

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

   8. Applied rewrites 65.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, -1\right), im, 1\right) + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot 0.5\\ t_1 := \left(e^{im} + e^{-im}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, -1\right), im, 1\right) + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) 0.5)) (t_1 (* (+ (exp im) (exp (- im))) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (*
         (fma
          (fma 0.001388888888888889 (* im im) 0.041666666666666664)
          (* im im)
          0.5)
         im)
        im
        1.0)
       (* (* re re) -0.16666666666666666))
      re)
     (if (<= t_1 5e+36)
       (* (fma im im 2.0) t_0)
       (* (+ (fma (fma im 0.5 -1.0) im 1.0) (exp im)) (* re 0.5))))))

C

double code(double re, double im) {
	double t_0 = sin(re) * 0.5;
	double t_1 = (exp(im) + exp(-im)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma((fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5) * im), im, 1.0) * ((re * re) * -0.16666666666666666)) * re;
	} else if (t_1 <= 5e+36) {
		tmp = fma(im, im, 2.0) * t_0;
	} else {
		tmp = (fma(fma(im, 0.5, -1.0), im, 1.0) + exp(im)) * (re * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * 0.5)
	t_1 = Float64(Float64(exp(im) + exp(Float64(-im))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5) * im), im, 1.0) * Float64(Float64(re * re) * -0.16666666666666666)) * re);
	elseif (t_1 <= 5e+36)
		tmp = Float64(fma(im, im, 2.0) * t_0);
	else
		tmp = Float64(Float64(fma(fma(im, 0.5, -1.0), im, 1.0) + exp(im)) * Float64(re * 0.5));
	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[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$1, 5e+36], N[(N[(im * im + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(im * 0.5 + -1.0), $MachinePrecision] * im + 1.0), $MachinePrecision] + N[Exp[im], $MachinePrecision]), $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

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 35.6%

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

   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))) < 4.99999999999999977e36

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 4.99999999999999977e36 < (*.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.3

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

   8. Applied rewrites 65.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, -1\right), im, 1\right) + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right)\\ t_1 := \sin re \cdot 0.5\\ t_2 := \left(e^{im} + e^{-im}\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (*
           (fma
            (fma 0.001388888888888889 (* im im) 0.041666666666666664)
            (* im im)
            0.5)
           im)
          im
          1.0))
        (t_1 (* (sin re) 0.5))
        (t_2 (* (+ (exp im) (exp (- im))) t_1)))
   (if (<= t_2 (- INFINITY))
     (* (* t_0 (* (* re re) -0.16666666666666666)) re)
     (if (<= t_2 5e+36) (* (fma im im 2.0) t_1) (* t_0 re)))))

C

double code(double re, double im) {
	double t_0 = fma((fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5) * im), im, 1.0);
	double t_1 = sin(re) * 0.5;
	double t_2 = (exp(im) + exp(-im)) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 * ((re * re) * -0.16666666666666666)) * re;
	} else if (t_2 <= 5e+36) {
		tmp = fma(im, im, 2.0) * t_1;
	} else {
		tmp = t_0 * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5) * im), im, 1.0)
	t_1 = Float64(sin(re) * 0.5)
	t_2 = Float64(Float64(exp(im) + exp(Float64(-im))) * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(Float64(re * re) * -0.16666666666666666)) * re);
	elseif (t_2 <= 5e+36)
		tmp = Float64(fma(im, im, 2.0) * t_1);
	else
		tmp = Float64(t_0 * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$2, 5e+36], N[(N[(im * im + 2.0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$0 * re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 35.6%

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

   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))) < 4.99999999999999977e36

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 4.99999999999999977e36 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 78.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right)\\ t_1 := \left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (*
           (fma
            (fma 0.001388888888888889 (* im im) 0.041666666666666664)
            (* im im)
            0.5)
           im)
          im
          1.0))
        (t_1 (* (+ (exp im) (exp (- im))) (* (sin re) 0.5))))
   (if (<= t_1 (- INFINITY))
     (* (* t_0 (* (* re re) -0.16666666666666666)) re)
     (if (<= t_1 5e+36) (sin re) (* t_0 re)))))

C

double code(double re, double im) {
	double t_0 = fma((fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5) * im), im, 1.0);
	double t_1 = (exp(im) + exp(-im)) * (sin(re) * 0.5);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_0 * ((re * re) * -0.16666666666666666)) * re;
	} else if (t_1 <= 5e+36) {
		tmp = sin(re);
	} else {
		tmp = t_0 * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5) * im), im, 1.0)
	t_1 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(Float64(re * re) * -0.16666666666666666)) * re);
	elseif (t_1 <= 5e+36)
		tmp = sin(re);
	else
		tmp = Float64(t_0 * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$1, 5e+36], N[Sin[re], $MachinePrecision], N[(t$95$0 * re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 35.6%

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

   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))) < 4.99999999999999977e36

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 4.99999999999999977e36 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 78.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right)\\ \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 0.002:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(t\_0, \left(\left(im \cdot im\right) \cdot im\right) \cdot im, \mathsf{fma}\left(im \cdot 0.5, im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_0, im \cdot im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma 0.001388888888888889 (* im im) 0.041666666666666664)))
   (if (<= (* (+ (exp im) (exp (- im))) (* (sin re) 0.5)) 0.002)
     (*
      (* (fma -0.16666666666666666 (* re re) 1.0) re)
      (fma t_0 (* (* (* im im) im) im) (fma (* im 0.5) im 1.0)))
     (* (fma (* (fma t_0 (* im im) 0.5) im) im 1.0) re))))

C

double code(double re, double im) {
	double t_0 = fma(0.001388888888888889, (im * im), 0.041666666666666664);
	double tmp;
	if (((exp(im) + exp(-im)) * (sin(re) * 0.5)) <= 0.002) {
		tmp = (fma(-0.16666666666666666, (re * re), 1.0) * re) * fma(t_0, (((im * im) * im) * im), fma((im * 0.5), im, 1.0));
	} else {
		tmp = fma((fma(t_0, (im * im), 0.5) * im), im, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(0.001388888888888889, Float64(im * im), 0.041666666666666664)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(sin(re) * 0.5)) <= 0.002)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re) * fma(t_0, Float64(Float64(Float64(im * im) * im) * im), fma(Float64(im * 0.5), im, 1.0)));
	else
		tmp = Float64(fma(Float64(fma(t_0, Float64(im * im), 0.5) * im), im, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(t$95$0 * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] + N[(N[(im * 0.5), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 61.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.2%

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

4. Final simplification 58.8%

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

Alternative 9: 57.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*l* N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.2%

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

4. Final simplification 55.8%

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

Alternative 10: 57.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*l* N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 38.8%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 36.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.0%

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

4. Final simplification 55.6%

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

Alternative 11: 53.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 61.8%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 54.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.2%

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

4. Final simplification 54.2%

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

Alternative 12: 52.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 61.8%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 54.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*l* N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 49.4%

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

4. Final simplification 52.4%

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

Alternative 13: 40.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 47.9%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 33.4%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 37.8%

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

4. Final simplification 40.6%

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

Alternative 14: 30.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((exp(im) + exp(-im)) * (sin(re) * 0.5d0)) <= (-0.002d0)) then
        tmp = ((re * re) * re) * (-0.16666666666666666d0)
    else
        tmp = 1.0d0 * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((Math.exp(im) + Math.exp(-im)) * (Math.sin(re) * 0.5)) <= -0.002) {
		tmp = ((re * re) * re) * -0.16666666666666666;
	} else {
		tmp = 1.0 * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((math.exp(im) + math.exp(-im)) * (math.sin(re) * 0.5)) <= -0.002:
		tmp = ((re * re) * re) * -0.16666666666666666
	else:
		tmp = 1.0 * re
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((exp(im) + exp(-im)) * (sin(re) * 0.5)) <= -0.002)
		tmp = ((re * re) * re) * -0.16666666666666666;
	else
		tmp = 1.0 * re;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 42.4

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

   5. Applied rewrites 42.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 15.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 14.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 54.7

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

   5. Applied rewrites 54.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.9%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 34.8%

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

4. Final simplification 26.7%

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

Alternative 15: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub0-neg N/A

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

   13. cosh-undef N/A

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

   14. associate-*r* N/A

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

   15. metadata-eval N/A

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

   16. exp-0 N/A

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

   17. lower-*.f64 N/A

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

   18. exp-0 N/A

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

   19. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 100.0

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

   4. lift-*.f64 N/A

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

   5. *-lft-identity 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 16: 58.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right)\\ \mathbf{if}\;\sin re \leq 0.002:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot t\_0\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (*
           (fma
            (fma 0.001388888888888889 (* im im) 0.041666666666666664)
            (* im im)
            0.5)
           im)
          im
          1.0)))
   (if (<= (sin re) 0.002)
     (* (* (fma -0.16666666666666666 (* re re) 1.0) t_0) re)
     (* t_0 re))))

C

double code(double re, double im) {
	double t_0 = fma((fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5) * im), im, 1.0);
	double tmp;
	if (sin(re) <= 0.002) {
		tmp = (fma(-0.16666666666666666, (re * re), 1.0) * t_0) * re;
	} else {
		tmp = t_0 * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5) * im), im, 1.0)
	tmp = 0.0
	if (sin(re) <= 0.002)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * t_0) * re);
	else
		tmp = Float64(t_0 * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision]}, If[LessEqual[N[Sin[re], $MachinePrecision], 0.002], N[(N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * re), $MachinePrecision], N[(t$95$0 * re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 30.1%

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

Alternative 17: 58.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right) \cdot im, im, 1\right)\\ \mathbf{if}\;\sin re \leq -0.002:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (*
           (fma
            (fma 0.001388888888888889 (* im im) 0.041666666666666664)
            (* im im)
            0.5)
           im)
          im
          1.0)))
   (if (<= (sin re) -0.002)
     (* (* t_0 (* (* re re) -0.16666666666666666)) re)
     (* t_0 re))))

C

double code(double re, double im) {
	double t_0 = fma((fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5) * im), im, 1.0);
	double tmp;
	if (sin(re) <= -0.002) {
		tmp = (t_0 * ((re * re) * -0.16666666666666666)) * re;
	} else {
		tmp = t_0 * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5) * im), im, 1.0)
	tmp = 0.0
	if (sin(re) <= -0.002)
		tmp = Float64(Float64(t_0 * Float64(Float64(re * re) * -0.16666666666666666)) * re);
	else
		tmp = Float64(t_0 * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision]}, If[LessEqual[N[Sin[re], $MachinePrecision], -0.002], N[(N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(t$95$0 * re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 29.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 71.1%

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

4. Final simplification 58.7%

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

Alternative 18: 86.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.155:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right) \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(im \cdot im\right) \cdot 0.041666666666666664\right) \cdot im\right) \cdot im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.155)
   (* (fma im im 2.0) (* (sin re) 0.5))
   (if (<= im 2.6e+77)
     (* (* (fma -0.16666666666666666 (* re re) 1.0) re) (cosh im))
     (* (* (* (* (* im im) 0.041666666666666664) im) im) (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.155) {
		tmp = fma(im, im, 2.0) * (sin(re) * 0.5);
	} else if (im <= 2.6e+77) {
		tmp = (fma(-0.16666666666666666, (re * re), 1.0) * re) * cosh(im);
	} else {
		tmp = ((((im * im) * 0.041666666666666664) * im) * im) * sin(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.155)
		tmp = Float64(fma(im, im, 2.0) * Float64(sin(re) * 0.5));
	elseif (im <= 2.6e+77)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re) * cosh(im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(im * im) * 0.041666666666666664) * im) * im) * sin(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.155], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.154999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if 0.154999999999999999 < im < 2.6000000000000002e77

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.9

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

   9. Applied rewrites 78.9%

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

   if 2.6000000000000002e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*l* N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.155:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right) \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(im \cdot im\right) \cdot 0.041666666666666664\right) \cdot im\right) \cdot im\right) \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 55.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.002)
		tmp = Float64(Float64(fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666) * re) * Float64(re * re));
	else
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.002], N[(N[(N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision] * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.7%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 27.2%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 27.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*l* N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 64.7%

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

Alternative 20: 48.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.002)
		tmp = Float64(Float64(fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666) * re) * Float64(re * re));
	else
		tmp = Float64(fma(Float64(im * im), 0.5, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.002], N[(N[(N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision] * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.7%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 27.2%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 27.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.8%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 54.2%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 56.5%

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

Alternative 21: 34.3% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-sin.f64 49.7

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

5. Applied rewrites 49.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 30.0%

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

Alternative 22: 26.2% accurate, 52.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-sin.f64 49.7

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

5. Applied rewrites 49.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 32.6%

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

9. Taylor expanded in re around 0

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

11. Applied rewrites 21.6%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))