math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.3s

Alternatives: 22

Speedup: 1.5×

• 48.0% 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, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub0-neg N/A

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

   13. cosh-undef N/A

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

   14. associate-*r* N/A

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

   15. metadata-eval N/A

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

   16. exp-0 N/A

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

   17. lower-*.f64 N/A

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

   18. exp-0 N/A

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

   19. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 85.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.0%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 67.0%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 66.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 74.5

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

   7. Applied rewrites 74.5%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 74.5

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

   9. Applied rewrites 74.5%

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

4. Final simplification 86.0%

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

Alternative 3: 83.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.0%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 67.0%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 66.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 68.9%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 68.9%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 68.9%

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

4. Final simplification 84.7%

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

Alternative 4: 83.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.0%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 67.0%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 66.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 99.0

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

   5. Applied rewrites 99.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 68.9%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 68.9%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 68.9%

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

4. Final simplification 84.3%

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

Alternative 5: 90.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 95.8

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

   7. Applied rewrites 95.8%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. cube-unmult N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 74.5

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

   7. Applied rewrites 74.5%

      \[\leadsto \left(1 \cdot \cosh im\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 74.5

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

   9. Applied rewrites 74.5%

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

4. Final simplification 91.5%

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

Alternative 6: 58.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right)\\ t_1 := im \cdot \left(im \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(t\_0, t\_1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right) \cdot \mathsf{fma}\left(t\_0, t\_1, \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma im (* im 0.001388888888888889) 0.041666666666666664))
        (t_1 (* im (* im (* im im)))))
   (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.1)
     (* (fma -0.16666666666666666 (* re (* re re)) re) (fma t_0 t_1 1.0))
     (*
      (fma
       (* (fma re (* re 0.008333333333333333) -0.16666666666666666) (* re re))
       re
       re)
      (fma t_0 t_1 (fma 0.5 (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = fma(im, (im * 0.001388888888888889), 0.041666666666666664);
	double t_1 = im * (im * (im * im));
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = fma(-0.16666666666666666, (re * (re * re)), re) * fma(t_0, t_1, 1.0);
	} else {
		tmp = fma((fma(re, (re * 0.008333333333333333), -0.16666666666666666) * (re * re)), re, re) * fma(t_0, t_1, fma(0.5, (im * im), 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664)
	t_1 = Float64(im * Float64(im * Float64(im * im)))
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * Float64(re * re)), re) * fma(t_0, t_1, 1.0));
	else
		tmp = Float64(fma(Float64(fma(re, Float64(re * 0.008333333333333333), -0.16666666666666666) * Float64(re * re)), re, re) * fma(t_0, t_1, fma(0.5, Float64(im * im), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(t$95$0 * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * N[(re * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision] * N[(t$95$0 * t$95$1 + N[(0.5 * N[(im * 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))) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 44.5%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 44.3%

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

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 71.5%

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

   10. Applied rewrites 71.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-152}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 5e-152)
   (*
    re
    (*
     (fma
      (* im im)
      (fma
       (* im im)
       (fma (* im im) 0.001388888888888889 0.041666666666666664)
       0.5)
      1.0)
     (fma (* re re) -0.16666666666666666 1.0)))
   (*
    (fma
     (fma re (* re 0.008333333333333333) -0.16666666666666666)
     (* re (* re re))
     re)
    (fma
     (fma im (* im 0.001388888888888889) 0.041666666666666664)
     (* im (* im (* im im)))
     1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 5e-152) {
		tmp = re * (fma((im * im), fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0) * fma((re * re), -0.16666666666666666, 1.0));
	} else {
		tmp = fma(fma(re, (re * 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re) * fma(fma(im, (im * 0.001388888888888889), 0.041666666666666664), (im * (im * (im * im))), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 5e-152)
		tmp = Float64(re * Float64(fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0) * fma(Float64(re * re), -0.16666666666666666, 1.0)));
	else
		tmp = Float64(fma(fma(re, Float64(re * 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re) * fma(fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), Float64(im * Float64(im * Float64(im * im))), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-152], N[(re * N[(N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * N[(re * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.9999999999999997e-152

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 63.3%

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

   if 4.9999999999999997e-152 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.0%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 54.0%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-152}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right)\\ \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, t\_0, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot t\_0, 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) 0.001388888888888889 0.041666666666666664)))
   (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 4e-5)
     (*
      re
      (*
       (fma (* im im) (fma (* im im) t_0 0.5) 1.0)
       (fma (* re re) -0.16666666666666666 1.0)))
     (fma re (* (* im im) (fma im (* im t_0) 0.5)) re))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), 0.001388888888888889, 0.041666666666666664);
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 4e-5) {
		tmp = re * (fma((im * im), fma((im * im), t_0, 0.5), 1.0) * fma((re * re), -0.16666666666666666, 1.0));
	} else {
		tmp = fma(re, ((im * im) * fma(im, (im * t_0), 0.5)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 4e-5)
		tmp = Float64(re * Float64(fma(Float64(im * im), fma(Float64(im * im), t_0, 0.5), 1.0) * fma(Float64(re * re), -0.16666666666666666, 1.0)));
	else
		tmp = fma(re, Float64(Float64(im * im) * fma(im, Float64(im * t_0), 0.5)), re);
	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[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], N[(re * N[(N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * t$95$0 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * t$95$0), $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))) < 4.00000000000000033e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 65.8%

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

   if 4.00000000000000033e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.7%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 45.8%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.1)
   (*
    (fma -0.16666666666666666 (* re (* re re)) re)
    (fma
     (fma im (* im 0.001388888888888889) 0.041666666666666664)
     (* im (* im (* im im)))
     1.0))
   (fma
    re
    (*
     (* im im)
     (fma
      im
      (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))
      0.5))
    re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = fma(-0.16666666666666666, (re * (re * re)), re) * fma(fma(im, (im * 0.001388888888888889), 0.041666666666666664), (im * (im * (im * im))), 1.0);
	} else {
		tmp = fma(re, ((im * im) * fma(im, (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)), 0.5)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * Float64(re * re)), re) * fma(fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), Float64(im * Float64(im * Float64(im * im))), 1.0));
	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[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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))) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 44.5%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 44.3%

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

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 70.8%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 4e-5)
   (*
    (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0)
    (fma re (* -0.16666666666666666 (* re re)) re))
   (fma
    re
    (*
     (* im im)
     (fma
      im
      (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))
      0.5))
    re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 4e-5) {
		tmp = fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0) * fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = fma(re, ((im * im) * fma(im, (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)), 0.5)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 4e-5)
		tmp = Float64(fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0) * fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re));
	else
		tmp = fma(re, Float64(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[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], N[(N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(re * N[(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))) < 4.00000000000000033e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 63.2%

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

   if 4.00000000000000033e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.7%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 45.8%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right) \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 4e-5)
   (* (fma re (* -0.16666666666666666 (* re re)) re) (fma 0.5 (* im im) 1.0))
   (fma
    re
    (*
     (* im im)
     (fma
      im
      (* im (fma (* im im) 0.001388888888888889 0.041666666666666664))
      0.5))
    re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 4e-5) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re) * fma(0.5, (im * im), 1.0);
	} else {
		tmp = fma(re, ((im * im) * fma(im, (im * fma((im * im), 0.001388888888888889, 0.041666666666666664)), 0.5)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 4e-5)
		tmp = Float64(fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re) * fma(0.5, Float64(im * im), 1.0));
	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[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], N[(N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $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))) < 4.00000000000000033e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 58.3%

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

   if 4.00000000000000033e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.7%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 45.8%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 4e-5)
   (* (fma re (* -0.16666666666666666 (* re re)) re) (fma 0.5 (* im im) 1.0))
   (fma (* im (* im (fma im (* im 0.041666666666666664) 0.5))) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 4e-5) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re) * fma(0.5, (im * im), 1.0);
	} else {
		tmp = fma((im * (im * fma(im, (im * 0.041666666666666664), 0.5))), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 4e-5)
		tmp = Float64(fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re) * fma(0.5, Float64(im * im), 1.0));
	else
		tmp = fma(Float64(im * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5))), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], N[(N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.00000000000000033e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 58.3%

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

   if 4.00000000000000033e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.6%

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

   10. Applied rewrites 44.5%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.1)
   (* (fma 0.5 (* im im) 1.0) (* re (* -0.16666666666666666 (* re re))))
   (fma (* im (* im (fma im (* im 0.041666666666666664) 0.5))) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = fma(0.5, (im * im), 1.0) * (re * (-0.16666666666666666 * (re * re)));
	} else {
		tmp = fma((im * (im * fma(im, (im * 0.041666666666666664), 0.5))), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(fma(0.5, Float64(im * im), 1.0) * Float64(re * Float64(-0.16666666666666666 * Float64(re * re))));
	else
		tmp = fma(Float64(im * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5))), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 31.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 15.1%

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

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 65.1%

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

   10. Applied rewrites 70.2%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.1)
   (* re (* re (* re -0.16666666666666666)))
   (fma (* im (* im (fma im (* im 0.041666666666666664) 0.5))) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = re * (re * (re * -0.16666666666666666));
	} else {
		tmp = fma((im * (im * fma(im, (im * 0.041666666666666664), 0.5))), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(re * Float64(re * Float64(re * -0.16666666666666666)));
	else
		tmp = fma(Float64(im * Float64(im * fma(im, Float64(im * 0.041666666666666664), 0.5))), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(im * N[(im * N[(im * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 36.3

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

   5. Applied rewrites 36.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 13.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 12.6%

       \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 12.6%

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

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 65.1%

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

   10. Applied rewrites 70.2%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot 0.041666666666666664, 0.5\right)\right), re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664, re \cdot \left(im \cdot im\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 4e-5)
   (fma re (* -0.16666666666666666 (* re re)) re)
   (fma (* (* im im) 0.041666666666666664) (* re (* im im)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 4e-5) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = fma(((im * im) * 0.041666666666666664), (re * (im * im)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 4e-5)
		tmp = fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re);
	else
		tmp = fma(Float64(Float64(im * im) * 0.041666666666666664), Float64(re * Float64(im * im)), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(re * 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))) < 4.00000000000000033e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 47.1%

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

   if 4.00000000000000033e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 34.6%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664, re \cdot \left(im \cdot im\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 4e-5)
   (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 (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 4e-5) {
		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(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 4e-5)
		tmp = fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re);
	else
		tmp = Float64(Float64(im * im) * Float64(re * fma(Float64(im * im), 0.041666666666666664, 0.5)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * 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))) < 4.00000000000000033e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 47.1%

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

   if 4.00000000000000033e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 34.7%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 0.4)
   (fma re (* -0.16666666666666666 (* re re)) re)
   (* (* im im) (* 0.041666666666666664 (* re (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 0.4) {
		tmp = fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = (im * im) * (0.041666666666666664 * (re * (im * im)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 0.4)
		tmp = fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re);
	else
		tmp = Float64(Float64(im * im) * Float64(0.041666666666666664 * Float64(re * Float64(im * im))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.4], N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 * N[(re * N[(im * im), $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))) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 62.9

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.9%

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

   if 0.40000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 36.8%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 36.9%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.041666666666666664 \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq 2 \cdot 10^{-37}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) 2e-37)
   (*
    re
    (*
     (fma
      (* im im)
      (fma
       (* im im)
       (fma (* im im) 0.001388888888888889 0.041666666666666664)
       0.5)
      1.0)
     (fma (* re re) -0.16666666666666666 1.0)))
   (*
    (fma
     (fma re (* re 0.008333333333333333) -0.16666666666666666)
     (* re (* re re))
     re)
    (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= 2e-37) {
		tmp = re * (fma((im * im), fma((im * im), fma((im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0) * fma((re * re), -0.16666666666666666, 1.0));
	} else {
		tmp = fma(fma(re, (re * 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= 2e-37)
		tmp = Float64(re * Float64(fma(Float64(im * im), fma(Float64(im * im), fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0) * fma(Float64(re * re), -0.16666666666666666, 1.0)));
	else
		tmp = Float64(fma(fma(re, Float64(re * 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], 2e-37], N[(re * N[(N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * N[(re * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 2.00000000000000013e-37

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.6%

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

   if 2.00000000000000013e-37 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.0%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 2 \cdot 10^{-37}:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot \mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 92.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{if}\;im \leq 0.115:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (sin re)
          (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0))))
   (if (<= im 0.115)
     t_0
     (if (<= im 2.5e+77)
       (* (cosh im) (fma re (* -0.16666666666666666 (* re re)) re))
       t_0))))

C

double code(double re, double im) {
	double t_0 = sin(re) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	double tmp;
	if (im <= 0.115) {
		tmp = t_0;
	} else if (im <= 2.5e+77) {
		tmp = cosh(im) * fma(re, (-0.16666666666666666 * (re * re)), re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0))
	tmp = 0.0
	if (im <= 0.115)
		tmp = t_0;
	elseif (im <= 2.5e+77)
		tmp = Float64(cosh(im) * fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.115], t$95$0, If[LessEqual[im, 2.5e+77], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.115000000000000005 or 2.50000000000000002e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

   5. Applied rewrites 94.9%

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

   if 0.115000000000000005 < im < 2.50000000000000002e77

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub0-neg N/A

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

      13. cosh-undef N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. exp-0 N/A

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

      17. lower-*.f64 N/A

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

      18. exp-0 N/A

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

      19. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.9

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

   7. Applied rewrites 76.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.115:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 47.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.005)
   (* re (* re (* re -0.16666666666666666)))
   (fma 0.5 (* re (* im im)) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.005) {
		tmp = re * (re * (re * -0.16666666666666666));
	} else {
		tmp = fma(0.5, (re * (im * im)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.005)
		tmp = Float64(re * Float64(re * Float64(re * -0.16666666666666666)));
	else
		tmp = fma(0.5, Float64(re * Float64(im * im)), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 18.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 17.9%

       \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 17.9%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 61.0%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 34.5% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-sin.f64 53.8

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

5. Applied rewrites 53.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 36.8%

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

9. Final simplification 36.8%

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

Alternative 22: 10.4% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (re * (re * (-0.16666666666666666d0)))
end function

Java

public static double code(double re, double im) {
	return re * (re * (re * -0.16666666666666666));
}

Python

def code(re, im):
	return re * (re * (re * -0.16666666666666666))

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = re * (re * (re * -0.16666666666666666));
end

Wolfram

code[re_, im_] := N[(re * N[(re * N[(re * -0.16666666666666666), $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. Taylor expanded in im around 0

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

   1. lower-sin.f64 53.8

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

5. Applied rewrites 53.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 36.8%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 10.5%

    \[\leadsto re \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
12. Step-by-step derivation

13. Applied rewrites 10.5%

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

14. Final simplification 10.5%

    \[\leadsto re \cdot \left(re \cdot \left(re \cdot -0.16666666666666666\right)\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))