math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 12.4s

Alternatives: 23

Speedup: 1.5×

• Could not converge on a ground truth

  1. re = 8.256814622029214e-267
  2. im = 7.288510389194822e+138

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. sub0-neg N/A

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

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

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. exp-lowering-exp.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 85.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 87.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

   8. Simplified 65.8%

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

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

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 99.2%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

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

   1. Initial program 100.0%

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

      1. flip3-+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}} \]

      2. clear-num N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{1}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

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

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

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

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

      7. clear-num N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \sin re}{\color{blue}{\frac{1}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}}}} \]

      8. flip3-+ N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \sin re}{\frac{1}{2 \cdot \cosh im}}} \]
   5. Step-by-step derivation

      1. frac-2neg N/A

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

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

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

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

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

      12. metadata-eval N/A

          \[\leadsto \frac{\sin re \cdot \frac{-1}{2}}{\frac{\color{blue}{\frac{-1}{2}}}{\cosh im}} \]

      13. cosh-lowering-cosh.f64 100.0

          \[\leadsto \frac{\sin re \cdot -0.5}{\frac{-0.5}{\color{blue}{\cosh im}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\sin re \cdot -0.5}{\frac{-0.5}{\cosh im}}} \]

   7. Taylor expanded in re around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot re}}{\frac{\frac{-1}{2}}{\cosh im}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{re \cdot \frac{-1}{2}}}{\frac{\frac{-1}{2}}{\cosh im}} \]

      2. *-lowering-*.f64 57.6

         \[\leadsto \frac{\color{blue}{re \cdot -0.5}}{\frac{-0.5}{\cosh im}} \]

   9. Simplified 57.6%

      \[\leadsto \frac{\color{blue}{re \cdot -0.5}}{\frac{-0.5}{\cosh im}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.7%

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

Alternative 3: 85.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 87.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

   8. Simplified 65.8%

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

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

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

      14. *-rgt-identity N/A

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

   5. Simplified 99.0%

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

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

   1. Initial program 100.0%

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

      1. flip3-+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}} \]

      2. clear-num N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{1}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

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

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

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

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

      7. clear-num N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \sin re}{\color{blue}{\frac{1}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}}}} \]

      8. flip3-+ N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \sin re}{\frac{1}{2 \cdot \cosh im}}} \]
   5. Step-by-step derivation

      1. frac-2neg N/A

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

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

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

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

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

      12. metadata-eval N/A

          \[\leadsto \frac{\sin re \cdot \frac{-1}{2}}{\frac{\color{blue}{\frac{-1}{2}}}{\cosh im}} \]

      13. cosh-lowering-cosh.f64 100.0

          \[\leadsto \frac{\sin re \cdot -0.5}{\frac{-0.5}{\color{blue}{\cosh im}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\sin re \cdot -0.5}{\frac{-0.5}{\cosh im}}} \]

   7. Taylor expanded in re around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot re}}{\frac{\frac{-1}{2}}{\cosh im}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{re \cdot \frac{-1}{2}}}{\frac{\frac{-1}{2}}{\cosh im}} \]

      2. *-lowering-*.f64 57.6

         \[\leadsto \frac{\color{blue}{re \cdot -0.5}}{\frac{-0.5}{\cosh im}} \]

   9. Simplified 57.6%

      \[\leadsto \frac{\color{blue}{re \cdot -0.5}}{\frac{-0.5}{\cosh im}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

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

Alternative 4: 83.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im))))
	t_1 = fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(im * im), fma(im, Float64(im * t_1), 0.5), 1.0)));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(re) * fma(Float64(im * im), fma(im, Float64(im * 0.041666666666666664), 0.5), 1.0));
	else
		tmp = Float64(fma(Float64(re * re), Float64(re * fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666)), re) * fma(t_1, Float64(im * Float64(im * Float64(im * im))), fma(im, Float64(0.5 * im), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * t$95$1), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(t$95$1 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im * N[(0.5 * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 87.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

   8. Simplified 65.8%

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

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

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

      14. *-rgt-identity N/A

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

   5. Simplified 99.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 79.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 50.3

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

   8. Simplified 50.3%

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

4. Final simplification 77.7%

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

Alternative 5: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ t_1 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right)\\ t_2 := \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot t\_2, 0.5\right), 1\right)\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin re \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(t\_2, im \cdot \left(im \cdot \left(im \cdot im\right)\right), t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma im (* 0.5 im) 1.0))
        (t_1 (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))))
        (t_2 (fma (* im im) 0.001388888888888889 0.041666666666666664)))
   (if (<= t_1 (- INFINITY))
     (*
      re
      (*
       (fma re (* re -0.16666666666666666) 1.0)
       (fma (* im im) (fma im (* im t_2) 0.5) 1.0)))
     (if (<= t_1 1.0)
       (* (sin re) t_0)
       (*
        (fma
         (* re re)
         (* re (fma (* re re) 0.008333333333333333 -0.16666666666666666))
         re)
        (fma t_2 (* im (* im (* im im))) t_0))))))

C

double code(double re, double im) {
	double t_0 = fma(im, (0.5 * im), 1.0);
	double t_1 = (0.5 * sin(re)) * (exp(im) + exp(-im));
	double t_2 = fma((im * im), 0.001388888888888889, 0.041666666666666664);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma((im * im), fma(im, (im * t_2), 0.5), 1.0));
	} else if (t_1 <= 1.0) {
		tmp = sin(re) * t_0;
	} else {
		tmp = fma((re * re), (re * fma((re * re), 0.008333333333333333, -0.16666666666666666)), re) * fma(t_2, (im * (im * (im * im))), t_0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(im, Float64(0.5 * im), 1.0)
	t_1 = Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im))))
	t_2 = fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(im * im), fma(im, Float64(im * t_2), 0.5), 1.0)));
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(re) * t_0);
	else
		tmp = Float64(fma(Float64(re * re), Float64(re * fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666)), re) * fma(t_2, Float64(im * Float64(im * Float64(im * im))), t_0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * im), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * t$95$2), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(t$95$2 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 87.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

   8. Simplified 65.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 98.8

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

   5. Simplified 98.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 79.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 50.3

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

   8. Simplified 50.3%

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

4. Final simplification 77.6%

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

Alternative 6: 82.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im))))
	t_1 = fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(im * im), fma(im, Float64(im * t_1), 0.5), 1.0)));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(fma(Float64(re * re), Float64(re * fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666)), re) * fma(t_1, Float64(im * Float64(im * Float64(im * im))), fma(im, Float64(0.5 * im), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * t$95$1), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(t$95$1 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im * N[(0.5 * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 87.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

   8. Simplified 65.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 98.1

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

   5. Simplified 98.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 79.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 50.3

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

   8. Simplified 50.3%

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

4. Final simplification 77.3%

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

Alternative 7: 58.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\\ \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.05:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot t\_0, 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(t\_0, im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) 0.001388888888888889 0.041666666666666664)))
   (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) -0.05)
     (*
      re
      (*
       (fma re (* re -0.16666666666666666) 1.0)
       (fma (* im im) (fma im (* im t_0) 0.5) 1.0)))
     (*
      (fma
       (* re re)
       (* re (fma (* re re) 0.008333333333333333 -0.16666666666666666))
       re)
      (fma t_0 (* im (* im (* im im))) (fma im (* 0.5 im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), 0.001388888888888889, 0.041666666666666664);
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= -0.05) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma((im * im), fma(im, (im * t_0), 0.5), 1.0));
	} else {
		tmp = fma((re * re), (re * fma((re * re), 0.008333333333333333, -0.16666666666666666)), re) * fma(t_0, (im * (im * (im * im))), fma(im, (0.5 * im), 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= -0.05)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(im * im), fma(im, Float64(im * t_0), 0.5), 1.0)));
	else
		tmp = Float64(fma(Float64(re * re), Float64(re * fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666)), re) * fma(t_0, Float64(im * Float64(im * Float64(im * im))), fma(im, Float64(0.5 * im), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]}, If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * t$95$0), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(t$95$0 * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im * N[(0.5 * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 90.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

   8. Simplified 49.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 91.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 61.0

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

   8. Simplified 61.0%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.05:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(0.08333333333333333, im \cdot im, 1\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 0.006)
   (*
    (fma (* im im) (fma 0.08333333333333333 (* im im) 1.0) 2.0)
    (* re (fma re (* re -0.08333333333333333) 0.5)))
   (fma re (* (* im im) (* im (* im (* im (* im 0.001388888888888889))))) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 0.006) {
		tmp = fma((im * im), fma(0.08333333333333333, (im * im), 1.0), 2.0) * (re * fma(re, (re * -0.08333333333333333), 0.5));
	} else {
		tmp = fma(re, ((im * im) * (im * (im * (im * (im * 0.001388888888888889))))), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 0.006)
		tmp = Float64(fma(Float64(im * im), fma(0.08333333333333333, Float64(im * im), 1.0), 2.0) * Float64(re * fma(re, Float64(re * -0.08333333333333333), 0.5)));
	else
		tmp = fma(re, Float64(Float64(im * im) * Float64(im * Float64(im * Float64(im * Float64(im * 0.001388888888888889))))), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.006], N[(N[(N[(im * im), $MachinePrecision] * N[(0.08333333333333333 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision] * N[(re * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0060000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

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

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

   8. Simplified 66.5%

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

   if 0.0060000000000000001 < (*.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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 86.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

   8. Simplified 34.5%

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

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

           \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(im \cdot \left(\frac{1}{720} \cdot \color{blue}{{im}^{3}}\right)\right), re\right) \]

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

           \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(\frac{1}{720} \cdot {im}^{3}\right)\right)}, re\right) \]

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

       18. unpow2 N/A

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

       19. associate-*l* N/A

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

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

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

       21. *-lowering-*.f64 34.5

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

   11. Simplified 34.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(0.08333333333333333, im \cdot im, 1\right), 2\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.05:\\ \;\;\;\;im \cdot \left(re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) -0.05)
   (*
    im
    (*
     re
     (*
      (fma (* re re) -0.16666666666666666 1.0)
      (* im (fma (* im im) 0.041666666666666664 0.5)))))
   (fma
    re
    (*
     (* im im)
     (fma
      im
      (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))
      0.5))
    re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= -0.05) {
		tmp = im * (re * (fma((re * re), -0.16666666666666666, 1.0) * (im * fma((im * im), 0.041666666666666664, 0.5))));
	} else {
		tmp = fma(re, ((im * im) * fma(im, (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)), 0.5)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= -0.05)
		tmp = Float64(im * Float64(re * Float64(fma(Float64(re * re), -0.16666666666666666, 1.0) * Float64(im * fma(Float64(im * im), 0.041666666666666664, 0.5)))));
	else
		tmp = fma(re, Float64(Float64(im * im) * fma(im, Float64(im * fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)), 0.5)), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(im * N[(re * N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 83.6

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

   5. Simplified 83.6%

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

   6. Taylor expanded in im around inf

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. metadata-eval N/A

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

      16. pow-sqr N/A

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

   8. Simplified 54.5%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

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

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. distribute-rgt1-in N/A

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

   11. Simplified 39.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 91.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

   8. Simplified 60.3%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.05:\\ \;\;\;\;im \cdot \left(re \cdot \left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.05:\\ \;\;\;\;im \cdot \left(im \cdot \left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right) \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) -0.05)
   (*
    im
    (*
     im
     (*
      (fma im (* im 0.041666666666666664) 0.5)
      (fma re (* -0.16666666666666666 (* re re)) re))))
   (fma
    re
    (*
     (* im im)
     (fma
      im
      (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))
      0.5))
    re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= -0.05) {
		tmp = im * (im * (fma(im, (im * 0.041666666666666664), 0.5) * fma(re, (-0.16666666666666666 * (re * re)), re)));
	} else {
		tmp = fma(re, ((im * im) * fma(im, (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)), 0.5)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= -0.05)
		tmp = Float64(im * Float64(im * Float64(fma(im, Float64(im * 0.041666666666666664), 0.5) * fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re))));
	else
		tmp = fma(re, Float64(Float64(im * im) * fma(im, Float64(im * fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)), 0.5)), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(im * N[(im * N[(N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 83.6

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

   5. Simplified 83.6%

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

   6. Taylor expanded in im around inf

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. metadata-eval N/A

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

      16. pow-sqr N/A

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

   8. Simplified 54.5%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

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

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

       5. *-commutative N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 37.9

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

   11. Simplified 37.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + {im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} \]
   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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 91.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

   8. Simplified 60.3%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -0.05:\\ \;\;\;\;im \cdot \left(im \cdot \left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right) \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 0.006)
   (* re (* (fma re (* re -0.16666666666666666) 1.0) (fma 0.5 (* im im) 1.0)))
   (fma re (* (* im im) (* im (* im (* im (* im 0.001388888888888889))))) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 0.006) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma(0.5, (im * im), 1.0));
	} else {
		tmp = fma(re, ((im * im) * (im * (im * (im * (im * 0.001388888888888889))))), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 0.006)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(0.5, Float64(im * im), 1.0)));
	else
		tmp = fma(re, Float64(Float64(im * im) * Float64(im * Float64(im * Float64(im * Float64(im * 0.001388888888888889))))), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.006], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(im * N[(im * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0060000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 79.3

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

   5. Simplified 79.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 61.6

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

   8. Simplified 61.6%

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

   if 0.0060000000000000001 < (*.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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 86.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

   8. Simplified 34.5%

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

   9. Taylor expanded in im around inf

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

           \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(im \cdot \left(\frac{1}{720} \cdot \color{blue}{{im}^{3}}\right)\right), re\right) \]

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

           \[\leadsto \mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(\frac{1}{720} \cdot {im}^{3}\right)\right)}, re\right) \]

       12. unpow3 N/A

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

       18. unpow2 N/A

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

       19. associate-*l* N/A

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

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

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

       21. *-lowering-*.f64 34.5

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

   11. Simplified 34.5%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot 0.001388888888888889\right)\right)\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 0.006)
   (* re (* (fma re (* re -0.16666666666666666) 1.0) (fma 0.5 (* im im) 1.0)))
   (fma re (* im (* im (fma (* im im) 0.041666666666666664 0.5))) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 0.006) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma(0.5, (im * im), 1.0));
	} else {
		tmp = fma(re, (im * (im * fma((im * im), 0.041666666666666664, 0.5))), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 0.006)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(0.5, Float64(im * im), 1.0)));
	else
		tmp = fma(re, Float64(im * Float64(im * fma(Float64(im * im), 0.041666666666666664, 0.5))), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.006], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0060000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 79.3

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

   5. Simplified 79.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 61.6

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

   8. Simplified 61.6%

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

   if 0.0060000000000000001 < (*.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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 86.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

   8. Simplified 34.5%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. associate-*r* N/A

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

       5. distribute-rgt-out N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

   11. Simplified 33.5%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 0.006)
   (fma re (* -0.16666666666666666 (* re re)) re)
   (fma re (* im (* im (fma (* im im) 0.041666666666666664 0.5))) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 0.006) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = fma(re, (im * (im * fma((im * im), 0.041666666666666664, 0.5))), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 0.006)
		tmp = fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re);
	else
		tmp = fma(re, Float64(im * Float64(im * fma(Float64(im * im), 0.041666666666666664, 0.5))), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.006], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0060000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 57.8

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

   5. Simplified 57.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 48.6

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

   8. Simplified 48.6%

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

   if 0.0060000000000000001 < (*.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-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 86.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

   8. Simplified 34.5%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. associate-*r* N/A

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

       5. distribute-rgt-out N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

   11. Simplified 33.5%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 0.006)
   (fma re (* -0.16666666666666666 (* re re)) re)
   (* re (* (* im im) (fma (* im im) 0.041666666666666664 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 0.006) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = re * ((im * im) * fma((im * im), 0.041666666666666664, 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 0.006)
		tmp = fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re);
	else
		tmp = Float64(re * Float64(Float64(im * im) * fma(Float64(im * im), 0.041666666666666664, 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.006], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0060000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. sin-lowering-sin.f64 57.8

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

   5. Simplified 57.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 48.6

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

   8. Simplified 48.6%

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

   if 0.0060000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 82.4

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

   5. Simplified 82.4%

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

   6. Taylor expanded in im around inf

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. metadata-eval N/A

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

      16. pow-sqr N/A

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

   8. Simplified 47.1%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

          \[\leadsto \color{blue}{re \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]

       6. unpow2 N/A

          \[\leadsto re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \]

       8. +-commutative N/A

          \[\leadsto re \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)}\right) \]

       9. *-commutative N/A

          \[\leadsto re \cdot \left(\left(im \cdot im\right) \cdot \left(\color{blue}{{im}^{2} \cdot \frac{1}{24}} + \frac{1}{2}\right)\right) \]

       10. accelerator-lowering-fma.f64 N/A

           \[\leadsto re \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{24}, \frac{1}{2}\right)}\right) \]

       11. unpow2 N/A

           \[\leadsto re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{24}, \frac{1}{2}\right)\right) \]

       12. *-lowering-*.f64 33.6

           \[\leadsto re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.041666666666666664, 0.5\right)\right) \]

   11. Simplified 33.6%

       \[\leadsto \color{blue}{re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 0.006)
   (fma re (* -0.16666666666666666 (* re re)) re)
   (* im (* im (* re (fma im (* im 0.041666666666666664) 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 0.006) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = im * (im * (re * fma(im, (im * 0.041666666666666664), 0.5)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 0.006)
		tmp = fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re);
	else
		tmp = Float64(im * Float64(im * Float64(re * fma(im, Float64(im * 0.041666666666666664), 0.5))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.006], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(im * N[(im * N[(re * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0060000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 57.8

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 57.8%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. *-lowering-*.f64 48.6

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   8. Simplified 48.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 0.0060000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{12}, im\right), 2\right) \]

      12. *-lowering-*.f64 82.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot 0.08333333333333333, im\right), 2\right) \]

   5. Simplified 82.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \sin re + \frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \sin re\right) + {im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right)} \]

      2. *-commutative N/A

         \[\leadsto {im}^{4} \cdot \left(\frac{1}{24} \cdot \sin re\right) + \color{blue}{\left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4}} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right) \cdot \sin re} + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      4. metadata-eval N/A

         \[\leadsto \left({im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \frac{1}{24}\right) \cdot \sin re + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      5. pow-sqr N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{1}{24}\right) \cdot \sin re + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \frac{1}{24}\right)\right)} \cdot \sin re + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \cdot \sin re + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      9. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)} \cdot \sin re\right) + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      10. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      12. associate-*r/ N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{{im}^{2}}} \cdot {im}^{4} \]

      13. associate-*l/ N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \color{blue}{\frac{\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{4}}{{im}^{2}}} \]

      14. associate-/l* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \frac{{im}^{4}}{{im}^{2}}} \]

      15. metadata-eval N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}} \]

      16. pow-sqr N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}} \]

   8. Simplified 47.1%

      \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\sin re \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(im \cdot \left(\color{blue}{re} \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 29.0%

       \[\leadsto im \cdot \left(im \cdot \left(\color{blue}{re} \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, 0.5 \cdot \left(im \cdot im\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 0.006)
   (fma re (* -0.16666666666666666 (* re re)) re)
   (fma re (* 0.5 (* im im)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 0.006) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = fma(re, (0.5 * (im * im)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 0.006)
		tmp = fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re);
	else
		tmp = fma(re, Float64(0.5 * Float64(im * im)), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.006], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0060000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 57.8

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 57.8%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \]

      8. *-lowering-*.f64 48.6

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot -0.16666666666666666, re\right) \]

   8. Simplified 48.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)} \]

   if 0.0060000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(im, \frac{1}{2} \cdot im, 1\right)} \]

      10. *-commutative N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{2}}, 1\right) \]

      11. *-lowering-*.f64 63.5

          \[\leadsto \sin re \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot 0.5}, 1\right) \]

   5. Simplified 63.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im, im \cdot 0.5, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{re} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2}, re\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{2} \cdot {im}^{2}}, re\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

      7. *-lowering-*.f64 24.6

         \[\leadsto \mathsf{fma}\left(re, 0.5 \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

   8. Simplified 24.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, 0.5 \cdot \left(im \cdot im\right), re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 0.006:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, 0.5 \cdot \left(im \cdot im\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp im) (exp (- im)))) 1.0)
   re
   (* re (* 0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 1.0) {
		tmp = re;
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((0.5d0 * sin(re)) * (exp(im) + exp(-im))) <= 1.0d0) then
        tmp = re
    else
        tmp = re * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.sin(re)) * (Math.exp(im) + Math.exp(-im))) <= 1.0) {
		tmp = re;
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.sin(re)) * (math.exp(im) + math.exp(-im))) <= 1.0:
		tmp = re
	else:
		tmp = re * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(im) + exp(Float64(-im)))) <= 1.0)
		tmp = re;
	else
		tmp = Float64(re * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * sin(re)) * (exp(im) + exp(-im))) <= 1.0)
		tmp = re;
	else
		tmp = re * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], re, N[(re * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 64.4

         \[\leadsto \color{blue}{\sin re} \]

   5. Simplified 64.4%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \]
   7. Step-by-step derivation

   8. Simplified 37.0%

      \[\leadsto \color{blue}{re} \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(im, \frac{1}{2} \cdot im, 1\right)} \]

      10. *-commutative N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{2}}, 1\right) \]

      11. *-lowering-*.f64 46.5

          \[\leadsto \sin re \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot 0.5}, 1\right) \]

   5. Simplified 46.5%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im, im \cdot 0.5, 1\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + re \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{re} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2}, re\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{2} \cdot {im}^{2}}, re\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

      7. *-lowering-*.f64 34.4

         \[\leadsto \mathsf{fma}\left(re, 0.5 \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

   8. Simplified 34.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, 0.5 \cdot \left(im \cdot im\right), re\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({im}^{2} \cdot re\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot re} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

       6. *-lowering-*.f64 34.4

          \[\leadsto re \cdot \left(0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   11. Simplified 34.4%

       \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sin re}{\frac{1}{\cosh im}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (sin re) (/ 1.0 (cosh im))))

C

double code(double re, double im) {
	return sin(re) / (1.0 / cosh(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) / (1.0d0 / cosh(im))
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) / (1.0 / Math.cosh(im));
}

Python

def code(re, im):
	return math.sin(re) / (1.0 / math.cosh(im))

Julia

function code(re, im)
	return Float64(sin(re) / Float64(1.0 / cosh(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) / (1.0 / cosh(im));
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] / N[(1.0 / N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip3-+ N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}} \]

   2. clear-num N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{1}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

   3. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

   7. clear-num N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \sin re}{\color{blue}{\frac{1}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}}}} \]

   8. flip3-+ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \sin re}{\frac{1}{\color{blue}{e^{0 - im} + e^{im}}}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sin re}{\frac{1}{2 \cdot \cosh im}}} \]
5. Step-by-step derivation

   1. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot \sin re\right)}{\mathsf{neg}\left(\frac{1}{2 \cdot \cosh im}\right)}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot \sin re\right)}{\mathsf{neg}\left(\frac{1}{2 \cdot \cosh im}\right)}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sin re \cdot \frac{1}{2}}\right)}{\mathsf{neg}\left(\frac{1}{2 \cdot \cosh im}\right)} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\color{blue}{\sin re \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\mathsf{neg}\left(\frac{1}{2 \cdot \cosh im}\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin re \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\mathsf{neg}\left(\frac{1}{2 \cdot \cosh im}\right)} \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin re} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{\mathsf{neg}\left(\frac{1}{2 \cdot \cosh im}\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\sin re \cdot \color{blue}{\frac{-1}{2}}}{\mathsf{neg}\left(\frac{1}{2 \cdot \cosh im}\right)} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\sin re \cdot \frac{-1}{2}}{\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2}}{\cosh im}}\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\sin re \cdot \frac{-1}{2}}{\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{\cosh im}\right)} \]

   10. distribute-neg-frac N/A

       \[\leadsto \frac{\sin re \cdot \frac{-1}{2}}{\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\cosh im}}} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \frac{\sin re \cdot \frac{-1}{2}}{\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\cosh im}}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\sin re \cdot \frac{-1}{2}}{\frac{\color{blue}{\frac{-1}{2}}}{\cosh im}} \]

   13. cosh-lowering-cosh.f64 100.0

       \[\leadsto \frac{\sin re \cdot -0.5}{\frac{-0.5}{\color{blue}{\cosh im}}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\sin re \cdot -0.5}{\frac{-0.5}{\cosh im}}} \]
7. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \frac{\sin re \cdot \frac{-1}{2}}{\color{blue}{\frac{-1}{2} \cdot \frac{1}{\cosh im}}} \]

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin re \cdot \frac{-1}{2}}{\frac{-1}{2}}}{\frac{1}{\cosh im}}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{\frac{-1}{2}}{\frac{-1}{2}}}}{\frac{1}{\cosh im}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\sin re \cdot \color{blue}{1}}{\frac{1}{\cosh im}} \]

   5. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{\sin re}}{\frac{1}{\cosh im}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin re}{\frac{1}{\cosh im}}} \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin re}}{\frac{1}{\cosh im}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \frac{\sin re}{\color{blue}{\frac{1}{\cosh im}}} \]

   9. cosh-lowering-cosh.f64 100.0

      \[\leadsto \frac{\sin re}{\frac{1}{\color{blue}{\cosh im}}} \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\sin re}{\frac{1}{\cosh im}}} \]
9. Add Preprocessing

Alternative 19: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \cosh im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sin re) (cosh im)))

C

double code(double re, double im) {
	return sin(re) * cosh(im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * cosh(im)
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) * Math.cosh(im);
}

Python

def code(re, im):
	return math.sin(re) * math.cosh(im)

Julia

function code(re, im)
	return Float64(sin(re) * cosh(im))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) * cosh(im);
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip3-+ N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}} \]

   2. clear-num N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\frac{1}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

   3. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\sin re}}{\frac{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}} \]

   7. clear-num N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \sin re}{\color{blue}{\frac{1}{\frac{{\left(e^{0 - im}\right)}^{3} + {\left(e^{im}\right)}^{3}}{e^{0 - im} \cdot e^{0 - im} + \left(e^{im} \cdot e^{im} - e^{0 - im} \cdot e^{im}\right)}}}} \]

   8. flip3-+ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \sin re}{\frac{1}{\color{blue}{e^{0 - im} + e^{im}}}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \sin re}{\frac{1}{2 \cdot \cosh im}}} \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin re \cdot \frac{1}{2}}}{\frac{1}{2 \cdot \cosh im}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\sin re \cdot \frac{\frac{1}{2}}{\frac{1}{2 \cdot \cosh im}}} \]

   3. clear-num N/A

      \[\leadsto \sin re \cdot \color{blue}{\frac{1}{\frac{\frac{1}{2 \cdot \cosh im}}{\frac{1}{2}}}} \]

   4. div-inv N/A

      \[\leadsto \sin re \cdot \frac{1}{\color{blue}{\frac{1}{2 \cdot \cosh im} \cdot \frac{1}{\frac{1}{2}}}} \]

   5. inv-pow N/A

      \[\leadsto \sin re \cdot \frac{1}{\color{blue}{{\left(2 \cdot \cosh im\right)}^{-1}} \cdot \frac{1}{\frac{1}{2}}} \]

   6. metadata-eval N/A

      \[\leadsto \sin re \cdot \frac{1}{{\left(2 \cdot \cosh im\right)}^{-1} \cdot \color{blue}{2}} \]

   7. metadata-eval N/A

      \[\leadsto \sin re \cdot \frac{1}{{\left(2 \cdot \cosh im\right)}^{-1} \cdot \color{blue}{{\frac{1}{2}}^{-1}}} \]

   8. unpow-prod-down N/A

      \[\leadsto \sin re \cdot \frac{1}{\color{blue}{{\left(\left(2 \cdot \cosh im\right) \cdot \frac{1}{2}\right)}^{-1}}} \]

   9. metadata-eval N/A

      \[\leadsto \sin re \cdot \frac{1}{{\left(\left(2 \cdot \cosh im\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{-1}} \]

   10. div-inv N/A

       \[\leadsto \sin re \cdot \frac{1}{{\color{blue}{\left(\frac{2 \cdot \cosh im}{2}\right)}}^{-1}} \]

   11. cosh-undef N/A

       \[\leadsto \sin re \cdot \frac{1}{{\left(\frac{\color{blue}{e^{im} + e^{\mathsf{neg}\left(im\right)}}}{2}\right)}^{-1}} \]

   12. cosh-def N/A

       \[\leadsto \sin re \cdot \frac{1}{{\color{blue}{\cosh im}}^{-1}} \]

   13. inv-pow N/A

       \[\leadsto \sin re \cdot \frac{1}{\color{blue}{\frac{1}{\cosh im}}} \]

   14. cosh-def N/A

       \[\leadsto \sin re \cdot \frac{1}{\frac{1}{\color{blue}{\frac{e^{im} + e^{\mathsf{neg}\left(im\right)}}{2}}}} \]

   15. cosh-undef N/A

       \[\leadsto \sin re \cdot \frac{1}{\frac{1}{\frac{\color{blue}{2 \cdot \cosh im}}{2}}} \]

   16. clear-num N/A

       \[\leadsto \sin re \cdot \frac{1}{\color{blue}{\frac{2}{2 \cdot \cosh im}}} \]

   17. clear-num N/A

       \[\leadsto \sin re \cdot \color{blue}{\frac{2 \cdot \cosh im}{2}} \]

   18. cosh-undef N/A

       \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im} + e^{\mathsf{neg}\left(im\right)}}}{2} \]

   19. cosh-def N/A

       \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
7. Add Preprocessing

Alternative 20: 58.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 4 \cdot 10^{-23}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(0.08333333333333333, im \cdot im, 1\right), 2\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 4e-23)
   (*
    re
    (*
     (fma re (* re -0.16666666666666666) 1.0)
     (fma
      (* im im)
      (fma
       im
       (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))
       0.5)
      1.0)))
   (*
    re
    (*
     (fma (* im im) (fma 0.08333333333333333 (* im im) 1.0) 2.0)
     (fma
      (* re re)
      (fma (* re re) 0.004166666666666667 -0.08333333333333333)
      0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 4e-23) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma((im * im), fma(im, (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0));
	} else {
		tmp = re * (fma((im * im), fma(0.08333333333333333, (im * im), 1.0), 2.0) * fma((re * re), fma((re * re), 0.004166666666666667, -0.08333333333333333), 0.5));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 4e-23)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0)));
	else
		tmp = Float64(re * Float64(fma(Float64(im * im), fma(0.08333333333333333, Float64(im * im), 1.0), 2.0) * fma(Float64(re * re), fma(Float64(re * re), 0.004166666666666667, -0.08333333333333333), 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 4e-23], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(N[(im * im), $MachinePrecision] * N[(0.08333333333333333 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.004166666666666667 + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 3.99999999999999984e-23

   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-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 92.4%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot 0.5, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 71.3%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)} \]

   if 3.99999999999999984e-23 < (sin.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{12}, im\right), 2\right) \]

      12. *-lowering-*.f64 84.3

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot 0.08333333333333333, im\right), 2\right) \]

   5. Simplified 84.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right) + {re}^{2} \cdot \left(\frac{-1}{12} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right) + \frac{1}{240} \cdot \left({re}^{2} \cdot \left(2 + im \cdot \left(im + \frac{1}{12} \cdot {im}^{3}\right)\right)\right)\right)\right)} \]

   8. Simplified 23.0%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(0.08333333333333333, im \cdot im, 1\right), 2\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 57.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 5 \cdot 10^{-8}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 5e-8)
   (*
    re
    (*
     (fma re (* re -0.16666666666666666) 1.0)
     (fma
      (* im im)
      (fma
       im
       (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))
       0.5)
      1.0)))
   (*
    im
    (*
     im
     (*
      (fma im (* im 0.041666666666666664) 0.5)
      (fma
       (fma (* re re) 0.008333333333333333 -0.16666666666666666)
       (* re (* re re))
       re))))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 5e-8) {
		tmp = re * (fma(re, (re * -0.16666666666666666), 1.0) * fma((im * im), fma(im, (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0));
	} else {
		tmp = im * (im * (fma(im, (im * 0.041666666666666664), 0.5) * fma(fma((re * re), 0.008333333333333333, -0.16666666666666666), (re * (re * re)), re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 5e-8)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(im * im), fma(im, Float64(im * fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0)));
	else
		tmp = Float64(im * Float64(im * Float64(fma(im, Float64(im * 0.041666666666666664), 0.5) * fma(fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666), Float64(re * Float64(re * re)), re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 5e-8], N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 4.9999999999999998e-8

   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-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right)} + \sin re \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}\right) + \sin re \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{720} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{24} \cdot \sin re\right)\right) + \left(\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin re\right)} \]

   5. Simplified 92.7%

      \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot 0.5, 1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(1 + \left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right) + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto re \cdot \left(1 + \color{blue}{\left(\left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \frac{-1}{6} \cdot \left({re}^{2} \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right) + \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]

      5. distribute-rgt1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(1 + \left(\frac{1}{2} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 72.4%

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)} \]

   if 4.9999999999999998e-8 < (sin.f64 re)

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} + 2\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right), 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{2} + 1\right)}, 2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + im \cdot 1}, 2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{1}{12} \cdot {im}^{2}\right) + \color{blue}{im}, 2\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{1}{12} \cdot {im}^{2}, im\right)}, 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{{im}^{2} \cdot \frac{1}{12}}, im\right), 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{1}{12}, im\right), 2\right) \]

      12. *-lowering-*.f64 82.8

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\left(im \cdot im\right)} \cdot 0.08333333333333333, im\right), 2\right) \]

   5. Simplified 82.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im, \left(im \cdot im\right) \cdot 0.08333333333333333, im\right), 2\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \sin re + \frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} \cdot \sin re\right) + {im}^{4} \cdot \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right)} \]

      2. *-commutative N/A

         \[\leadsto {im}^{4} \cdot \left(\frac{1}{24} \cdot \sin re\right) + \color{blue}{\left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4}} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{4} \cdot \frac{1}{24}\right) \cdot \sin re} + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      4. metadata-eval N/A

         \[\leadsto \left({im}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \frac{1}{24}\right) \cdot \sin re + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      5. pow-sqr N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot {im}^{2}\right)} \cdot \frac{1}{24}\right) \cdot \sin re + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \frac{1}{24}\right)\right)} \cdot \sin re + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \cdot \sin re + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re\right)} + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      9. *-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{1}{24}\right)} \cdot \sin re\right) + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      10. associate-*r* N/A

          \[\leadsto {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right)} + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2}} + \left(\frac{1}{2} \cdot \frac{\sin re}{{im}^{2}}\right) \cdot {im}^{4} \]

      12. associate-*r/ N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \color{blue}{\frac{\frac{1}{2} \cdot \sin re}{{im}^{2}}} \cdot {im}^{4} \]

      13. associate-*l/ N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \color{blue}{\frac{\left(\frac{1}{2} \cdot \sin re\right) \cdot {im}^{4}}{{im}^{2}}} \]

      14. associate-/l* N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \frac{{im}^{4}}{{im}^{2}}} \]

      15. metadata-eval N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot \frac{{im}^{\color{blue}{\left(2 \cdot 2\right)}}}{{im}^{2}} \]

      16. pow-sqr N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{24} \cdot \sin re\right)\right) \cdot {im}^{2} + \left(\frac{1}{2} \cdot \sin re\right) \cdot \frac{\color{blue}{{im}^{2} \cdot {im}^{2}}}{{im}^{2}} \]

   8. Simplified 39.7%

      \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(\sin re \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(im \cdot \left(\color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \left(im \cdot \left(\left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       2. distribute-lft-in N/A

          \[\leadsto im \cdot \left(im \cdot \left(\color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto im \cdot \left(im \cdot \left(\left(\color{blue}{\left(re \cdot {re}^{2}\right) \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)} + re \cdot 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto im \cdot \left(im \cdot \left(\left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + re \cdot 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       5. cube-mult N/A

          \[\leadsto im \cdot \left(im \cdot \left(\left(\color{blue}{{re}^{3}} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + re \cdot 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto im \cdot \left(im \cdot \left(\left(\color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{3}} + re \cdot 1\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       7. *-rgt-identity N/A

          \[\leadsto im \cdot \left(im \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{3} + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(im \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)} \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       9. sub-neg N/A

          \[\leadsto im \cdot \left(im \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto im \cdot \left(im \cdot \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto im \cdot \left(im \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(im \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto im \cdot \left(im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \left(im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       15. cube-mult N/A

           \[\leadsto im \cdot \left(im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto im \cdot \left(im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{{re}^{2}}, re\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \left(im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot {re}^{2}}, re\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto im \cdot \left(im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im, im \cdot \frac{1}{24}, \frac{1}{2}\right)\right)\right) \]

       19. *-lowering-*.f64 14.2

           \[\leadsto im \cdot \left(im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\right) \]

   11. Simplified 14.2%

       \[\leadsto im \cdot \left(im \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 5 \cdot 10^{-8}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(\mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 48.2% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(re, 0.5 \cdot \left(im \cdot im\right), re\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (fma re (* 0.5 (* im im)) re))

C

double code(double re, double im) {
	return fma(re, (0.5 * (im * im)), re);
}

Julia

function code(re, im)
	return fma(re, Float64(0.5 * Float64(im * im)), re)
end

Wolfram

code[re_, im_] := N[(re * N[(0.5 * N[(im * im), $MachinePrecision]), $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

   \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \sin re + \color{blue}{\left(\frac{1}{2} \cdot {im}^{2}\right) \cdot \sin re} \]

   2. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   3. unpow2 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)} + 1\right) \cdot \sin re \]

   4. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot im} + 1\right) \cdot \sin re \]

   5. *-commutative N/A

      \[\leadsto \left(\color{blue}{im \cdot \left(\frac{1}{2} \cdot im\right)} + 1\right) \cdot \sin re \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right)} \]

   8. sin-lowering-sin.f64 N/A

      \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right) + 1\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(im, \frac{1}{2} \cdot im, 1\right)} \]

   10. *-commutative N/A

       \[\leadsto \sin re \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{1}{2}}, 1\right) \]

   11. *-lowering-*.f64 73.3

       \[\leadsto \sin re \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot 0.5}, 1\right) \]

5. Simplified 73.3%

   \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(im, im \cdot 0.5, 1\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot {im}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot {im}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + re \cdot 1} \]

   3. *-rgt-identity N/A

      \[\leadsto re \cdot \left(\frac{1}{2} \cdot {im}^{2}\right) + \color{blue}{re} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \frac{1}{2} \cdot {im}^{2}, re\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{1}{2} \cdot {im}^{2}}, re\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(re, \frac{1}{2} \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

   7. *-lowering-*.f64 47.7

      \[\leadsto \mathsf{fma}\left(re, 0.5 \cdot \color{blue}{\left(im \cdot im\right)}, re\right) \]

8. Simplified 47.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(re, 0.5 \cdot \left(im \cdot im\right), re\right)} \]
9. Add Preprocessing

Alternative 23: 26.6% accurate, 317.0× speedup

Math

\[\begin{array}{l} \\ re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 re)

C

double code(double re, double im) {
	return re;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re
end function

Java

public static double code(double re, double im) {
	return re;
}

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

function tmp = code(re, im)
	tmp = re;
end

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 48.6

      \[\leadsto \color{blue}{\sin re} \]

5. Simplified 48.6%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re} \]
7. Step-by-step derivation

8. Simplified 27.9%

   \[\leadsto \color{blue}{re} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))