math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.1s

Alternatives: 20

Speedup: 1.5×

• 50.1% 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: 83.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (cosh im) (fma re (* (* re re) -0.16666666666666666) re))
     (if (<= t_0 1.0)
       (sin 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 t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = cosh(im) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} 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)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cosh(im) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = fma(re, Float64(Float64(im * im) * fma(Float64(im * im), fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), 0.5)), re);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. 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 72.3

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

   7. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 72.3

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

   9. Applied rewrites 72.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 73.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 60.6%

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

4. Final simplification 83.0%

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

Alternative 3: 77.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma 0.5 (* im im) 1.0)
      (fma
       (fma
        (* re re)
        (fma (* re re) -0.0001984126984126984 0.008333333333333333)
        -0.16666666666666666)
       (* re (* re re))
       re))
     (if (<= t_0 1.0)
       (sin 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 t_0 = (sin(re) * 0.5) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(0.5, (im * im), 1.0) * fma(fma((re * re), fma((re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re);
	} else if (t_0 <= 1.0) {
		tmp = sin(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)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(0.5, Float64(im * im), 1.0) * fma(fma(Float64(re * re), fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = fma(re, Float64(Float64(im * im) * fma(Float64(im * im), fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), 0.5)), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   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 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 55.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 73.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 60.6%

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

4. Final simplification 78.8%

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

Alternative 4: 53.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 2e-5)
   (*
    (fma 0.5 (* im im) 1.0)
    (fma
     (fma
      (* re re)
      (fma (* re re) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     (* re (* 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))) <= 2e-5) {
		tmp = fma(0.5, (im * im), 1.0) * fma(fma((re * re), fma((re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (re * (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))) <= 2e-5)
		tmp = Float64(fma(0.5, Float64(im * im), 1.0) * fma(fma(Float64(re * re), fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re));
	else
		tmp = fma(re, Float64(Float64(im * im) * fma(Float64(im * im), fma(im, Float64(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], 2e-5], N[(N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $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))) < 2.00000000000000016e-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 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 61.6%

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

   if 2.00000000000000016e-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 81.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.7%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 2e-5)
   (*
    (fma re (* (* re re) -0.16666666666666666) re)
    (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 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))) <= 2e-5) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 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))) <= 2e-5)
		tmp = Float64(fma(re, Float64(Float64(re * re) * -0.16666666666666666), re) * fma(Float64(im * im), fma(Float64(im * im), 0.041666666666666664, 0.5), 1.0));
	else
		tmp = fma(re, Float64(Float64(im * im) * fma(Float64(im * im), fma(im, Float64(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], 2e-5], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $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))) < 2.00000000000000016e-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 87.8%

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

   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 64.0%

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

   if 2.00000000000000016e-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 81.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.7%

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

4. Final simplification 56.7%

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

Alternative 6: 46.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \left(re \cdot re\right) \cdot -0.0001984126984126984, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 2e-5)
   (fma
    (fma (* re re) (* (* re re) -0.0001984126984126984) -0.16666666666666666)
    (* re (* 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))) <= 2e-5) {
		tmp = fma(fma((re * re), ((re * re) * -0.0001984126984126984), -0.16666666666666666), (re * (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))) <= 2e-5)
		tmp = fma(fma(Float64(re * re), Float64(Float64(re * re) * -0.0001984126984126984), -0.16666666666666666), Float64(re * Float64(re * re)), re);
	else
		tmp = fma(re, Float64(Float64(im * im) * fma(Float64(im * im), fma(im, Float64(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], 2e-5], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $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))) < 2.00000000000000016e-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 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.2%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{5040} \cdot {re}^{2}, \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.2%

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

   if 2.00000000000000016e-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 81.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.7%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \left(re \cdot re\right) \cdot -0.0001984126984126984, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \left(re \cdot re\right) \cdot -0.0001984126984126984, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 2e-5)
   (fma
    (fma (* re re) (* (* re re) -0.0001984126984126984) -0.16666666666666666)
    (* re (* re re))
    re)
   (fma (* im im) (* re (fma (* im im) 0.041666666666666664 0.5)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 2e-5) {
		tmp = fma(fma((re * re), ((re * re) * -0.0001984126984126984), -0.16666666666666666), (re * (re * re)), re);
	} else {
		tmp = fma((im * im), (re * fma((im * im), 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))) <= 2e-5)
		tmp = fma(fma(Float64(re * re), Float64(Float64(re * re) * -0.0001984126984126984), -0.16666666666666666), Float64(re * Float64(re * re)), re);
	else
		tmp = fma(Float64(im * im), Float64(re * fma(Float64(im * im), 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], 2e-5], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * 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] + 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))) < 2.00000000000000016e-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 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.2%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{-1}{5040} \cdot {re}^{2}, \frac{-1}{6}\right), re \cdot \left(re \cdot re\right), re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.2%

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

   if 2.00000000000000016e-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 79.1%

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

   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 37.5%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \left(re \cdot re\right) \cdot -0.0001984126984126984, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) -0.05)
   (fma (* re (* re (* (* re re) -0.0001984126984126984))) (* re (* re re)) re)
   (fma (* im im) (* re (fma (* im im) 0.041666666666666664 0.5)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= -0.05) {
		tmp = fma((re * (re * ((re * re) * -0.0001984126984126984))), (re * (re * re)), re);
	} else {
		tmp = fma((im * im), (re * fma((im * im), 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))) <= -0.05)
		tmp = fma(Float64(re * Float64(re * Float64(Float64(re * re) * -0.0001984126984126984))), Float64(re * Float64(re * re)), re);
	else
		tmp = fma(Float64(im * im), Float64(re * fma(Float64(im * im), 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], -0.05], N[(N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * 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] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-sin.f64 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.2%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{5040} \cdot {re}^{4}, re \cdot \left(\color{blue}{re} \cdot re\right), re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 23.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-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 87.7%

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

   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 63.0%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.0001984126984126984\right)\right), re \cdot \left(re \cdot re\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 2e-5)
   (* re (* (fma 0.5 (* im im) 1.0) (fma (* re re) -0.16666666666666666 1.0)))
   (fma (* im im) (* re (fma (* im im) 0.041666666666666664 0.5)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 2e-5) {
		tmp = re * (fma(0.5, (im * im), 1.0) * fma((re * re), -0.16666666666666666, 1.0));
	} else {
		tmp = fma((im * im), (re * fma((im * im), 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))) <= 2e-5)
		tmp = Float64(re * Float64(fma(0.5, Float64(im * im), 1.0) * fma(Float64(re * re), -0.16666666666666666, 1.0)));
	else
		tmp = fma(Float64(im * im), Float64(re * fma(Float64(im * im), 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], 2e-5], N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 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))) < 2.00000000000000016e-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 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 59.4%

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

   if 2.00000000000000016e-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 79.1%

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

   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 37.5%

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

4. Final simplification 51.6%

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

Alternative 10: 40.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 2e-5)
   (fma re (* (* re re) -0.16666666666666666) re)
   (fma 0.5 (* re (* im im)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 2e-5) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = fma(0.5, (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))) <= 2e-5)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = fma(0.5, 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], 2e-5], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision], N[(0.5 * 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))) < 2.00000000000000016e-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 61.5

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

   5. Applied rewrites 61.5%

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

   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 48.1%

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

   if 2.00000000000000016e-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

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   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 63.5

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

   5. Applied rewrites 63.5%

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

   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 28.1%

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

4. Final simplification 40.9%

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

Alternative 11: 40.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 2e-5)
   (fma re (* (* re re) -0.16666666666666666) re)
   (* re (* 0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 2e-5) {
		tmp = fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 2e-5)
		tmp = fma(re, Float64(Float64(re * re) * -0.16666666666666666), re);
	else
		tmp = Float64(re * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-5], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision], N[(re * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-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 61.5

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

   5. Applied rewrites 61.5%

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

   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 48.1%

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

   if 2.00000000000000016e-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

      \[\leadsto \color{blue}{\sin re + \frac{1}{2} \cdot \left({im}^{2} \cdot \sin re\right)} \]
   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 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 36.2%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 28.3%

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

4. Final simplification 41.0%

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

Alternative 12: 17.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-305}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* (sin re) 0.5) (+ (exp (- im)) (exp im))) 5e-305)
   (* -0.16666666666666666 (* re (* re re)))
   (* re (* 0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 5e-305) {
		tmp = -0.16666666666666666 * (re * (re * re));
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((sin(re) * 0.5d0) * (exp(-im) + exp(im))) <= 5d-305) then
        tmp = (-0.16666666666666666d0) * (re * (re * re))
    else
        tmp = re * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((Math.sin(re) * 0.5) * (Math.exp(-im) + Math.exp(im))) <= 5e-305) {
		tmp = -0.16666666666666666 * (re * (re * re));
	} else {
		tmp = re * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((math.sin(re) * 0.5) * (math.exp(-im) + math.exp(im))) <= 5e-305:
		tmp = -0.16666666666666666 * (re * (re * re))
	else:
		tmp = re * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im)) + exp(im))) <= 5e-305)
		tmp = Float64(-0.16666666666666666 * Float64(re * Float64(re * re)));
	else
		tmp = Float64(re * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((sin(re) * 0.5) * (exp(-im) + exp(im))) <= 5e-305)
		tmp = -0.16666666666666666 * (re * (re * re));
	else
		tmp = re * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-305], N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.99999999999999985e-305

   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 53.9

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

   5. Applied rewrites 53.9%

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

   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 37.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 12.2%

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

   if 4.99999999999999985e-305 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. 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 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 50.8%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.2%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 23.0%

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

4. Final simplification 17.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{-305}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\sin re \leq 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.02)
   (*
    re
    (* (* re re) (fma (* im im) -0.08333333333333333 -0.16666666666666666)))
   (if (<= (sin re) 1e-16)
     (fma 0.5 (* re (* im im)) re)
     (fma
      (fma re (* re 0.008333333333333333) -0.16666666666666666)
      (* re (* re re))
      re))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = re * ((re * re) * fma((im * im), -0.08333333333333333, -0.16666666666666666));
	} else if (sin(re) <= 1e-16) {
		tmp = fma(0.5, (re * (im * im)), re);
	} else {
		tmp = fma(fma(re, (re * 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(re * Float64(Float64(re * re) * fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666)));
	elseif (sin(re) <= 1e-16)
		tmp = fma(0.5, Float64(re * Float64(im * im)), re);
	else
		tmp = 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[Sin[re], $MachinePrecision], -0.02], N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[re], $MachinePrecision], 1e-16], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(N[(re * N[(re * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.5%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 26.5%

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

   if -0.0200000000000000004 < (sin.f64 re) < 9.9999999999999998e-17

   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 77.6

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

   5. Applied rewrites 77.6%

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

   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 77.4%

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

   if 9.9999999999999998e-17 < (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

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

      1. lower-sin.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 21.5%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\sin re \leq 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re, re \cdot 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\sin re \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.008333333333333333, re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.02)
   (*
    re
    (* (* re re) (fma (* im im) -0.08333333333333333 -0.16666666666666666)))
   (if (<= (sin re) 2e-5)
     (fma 0.5 (* re (* im im)) re)
     (fma (* (* re re) 0.008333333333333333) (* re (* re re)) re))))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = re * ((re * re) * fma((im * im), -0.08333333333333333, -0.16666666666666666));
	} else if (sin(re) <= 2e-5) {
		tmp = fma(0.5, (re * (im * im)), re);
	} else {
		tmp = fma(((re * re) * 0.008333333333333333), (re * (re * re)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(re * Float64(Float64(re * re) * fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666)));
	elseif (sin(re) <= 2e-5)
		tmp = fma(0.5, Float64(re * Float64(im * im)), re);
	else
		tmp = fma(Float64(Float64(re * re) * 0.008333333333333333), Float64(re * Float64(re * re)), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[re], $MachinePrecision], 2e-5], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.5%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 26.5%

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

   if -0.0200000000000000004 < (sin.f64 re) < 2.00000000000000016e-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 78.0

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

   5. Applied rewrites 78.0%

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

   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 77.8%

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

   if 2.00000000000000016e-5 < (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

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

      1. lower-sin.f64 49.0

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

   5. Applied rewrites 49.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 18.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 18.7%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\sin re \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.008333333333333333, re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot re\right)\\ \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;-0.08333333333333333 \cdot \left(im \cdot \left(im \cdot t\_0\right)\right)\\ \mathbf{elif}\;\sin re \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.008333333333333333, t\_0, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* re re))))
   (if (<= (sin re) -0.02)
     (* -0.08333333333333333 (* im (* im t_0)))
     (if (<= (sin re) 2e-5)
       (fma 0.5 (* re (* im im)) re)
       (fma (* (* re re) 0.008333333333333333) t_0 re)))))

C

double code(double re, double im) {
	double t_0 = re * (re * re);
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = -0.08333333333333333 * (im * (im * t_0));
	} else if (sin(re) <= 2e-5) {
		tmp = fma(0.5, (re * (im * im)), re);
	} else {
		tmp = fma(((re * re) * 0.008333333333333333), t_0, re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(re * Float64(re * re))
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(-0.08333333333333333 * Float64(im * Float64(im * t_0)));
	elseif (sin(re) <= 2e-5)
		tmp = fma(0.5, Float64(re * Float64(im * im)), re);
	else
		tmp = fma(Float64(Float64(re * re) * 0.008333333333333333), t_0, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(-0.08333333333333333 * N[(im * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[re], $MachinePrecision], 2e-5], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * t$95$0 + re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 26.1%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 26.4%

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

   if -0.0200000000000000004 < (sin.f64 re) < 2.00000000000000016e-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 78.0

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

   5. Applied rewrites 78.0%

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

   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 77.8%

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

   if 2.00000000000000016e-5 < (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

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

      1. lower-sin.f64 49.0

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

   5. Applied rewrites 49.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 18.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 18.7%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;-0.08333333333333333 \cdot \left(im \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{elif}\;\sin re \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.008333333333333333, re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.001388888888888889, 0.041666666666666664\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(0.5, im \cdot im, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e-10)
   (sin re)
   (if (<= im 7.2e+51)
     (* (cosh im) (fma re (* (* re re) -0.16666666666666666) re))
     (*
      (sin re)
      (fma
       (fma im (* im 0.001388888888888889) 0.041666666666666664)
       (* im (* im (* im im)))
       (fma 0.5 (* im im) 1.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e-10) {
		tmp = sin(re);
	} else if (im <= 7.2e+51) {
		tmp = cosh(im) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = sin(re) * fma(fma(im, (im * 0.001388888888888889), 0.041666666666666664), (im * (im * (im * im))), fma(0.5, (im * im), 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e-10)
		tmp = sin(re);
	elseif (im <= 7.2e+51)
		tmp = Float64(cosh(im) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	else
		tmp = Float64(sin(re) * fma(fma(im, Float64(im * 0.001388888888888889), 0.041666666666666664), Float64(im * Float64(im * Float64(im * im))), fma(0.5, Float64(im * im), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e-10], N[Sin[re], $MachinePrecision], If[LessEqual[im, 7.2e+51], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(im * N[(im * 0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * N[(im * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.1e-10

   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 69.1

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

   5. Applied rewrites 69.1%

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

   if 2.1e-10 < im < 7.20000000000000022e51

   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 68.4

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

   7. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 68.4

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

   9. Applied rewrites 68.4%

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

   if 7.20000000000000022e51 < 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 100.0%

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

Alternative 17: 72.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+77}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e-10)
   (sin re)
   (if (<= im 1.85e+77)
     (* (cosh im) (fma re (* (* re re) -0.16666666666666666) re))
     (*
      (sin re)
      (fma (* im im) (fma (* im im) 0.041666666666666664 0.5) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e-10) {
		tmp = sin(re);
	} else if (im <= 1.85e+77) {
		tmp = cosh(im) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else {
		tmp = sin(re) * fma((im * im), fma((im * im), 0.041666666666666664, 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e-10)
		tmp = sin(re);
	elseif (im <= 1.85e+77)
		tmp = Float64(cosh(im) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	else
		tmp = Float64(sin(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[im, 2.1e-10], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.85e+77], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[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 3 regimes

2. if im < 2.1e-10

   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 69.1

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

   5. Applied rewrites 69.1%

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

   if 2.1e-10 < im < 1.84999999999999997e77

   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 69.6

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

   7. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 69.6

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

   9. Applied rewrites 69.6%

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

   if 1.84999999999999997e77 < 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 98.3%

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

Alternative 18: 52.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.02)
   (*
    re
    (* (* re re) (fma (* im im) -0.08333333333333333 -0.16666666666666666)))
   (fma (* im im) (* re (fma (* im im) 0.041666666666666664 0.5)) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = re * ((re * re) * fma((im * im), -0.08333333333333333, -0.16666666666666666));
	} else {
		tmp = fma((im * im), (re * fma((im * im), 0.041666666666666664, 0.5)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(re * Float64(Float64(re * re) * fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666)));
	else
		tmp = fma(Float64(im * im), Float64(re * fma(Float64(im * im), 0.041666666666666664, 0.5)), re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.5%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 26.5%

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

   if -0.0200000000000000004 < (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 86.1%

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

   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 62.6%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, re \cdot \mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 48.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;-0.08333333333333333 \cdot \left(im \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin re) -0.02)
   (* -0.08333333333333333 (* im (* im (* re (* re re)))))
   (fma 0.5 (* re (* im im)) re)))

C

double code(double re, double im) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = -0.08333333333333333 * (im * (im * (re * (re * re))));
	} else {
		tmp = fma(0.5, (re * (im * im)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(-0.08333333333333333 * Float64(im * Float64(im * Float64(re * Float64(re * re)))));
	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], -0.02], N[(-0.08333333333333333 * N[(im * N[(im * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $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) < -0.0200000000000000004

   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 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 26.1%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 26.4%

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

   if -0.0200000000000000004 < (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 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 57.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;-0.08333333333333333 \cdot \left(im \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re \cdot \left(im \cdot im\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 10.7% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(-0.16666666666666666 * N[(re * N[(re * re), $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 50.6

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

5. Applied rewrites 50.6%

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

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 35.8%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 11.9%

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

Reproduce

herbie shell --seed 2024237 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))