math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.6s

Alternatives: 20

Speedup: 1.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (fma (* 0.5 (sin re)) (exp (- im_m)) (* (sin re) (* 0.5 (exp im_m)))))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[Exp[(-im$95$m)], $MachinePrecision] + N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub0-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 84.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := t\_0 \cdot \left(e^{im\_m} + e^{-im\_m}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+290}:\\ \;\;\;\;\cosh im\_m \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+22}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im\_m, \mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.002777777777777778, 0.08333333333333333\right), im\_m \cdot \left(im\_m \cdot im\_m\right), im\_m\right), 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* t_0 (+ (exp im_m) (exp (- im_m))))))
   (if (<= t_1 -5e+290)
     (* (cosh im_m) (fma re (* (* re re) -0.16666666666666666) re))
     (if (<= t_1 5e+22)
       (* t_0 (fma im_m im_m 2.0))
       (*
        (* 0.5 re)
        (fma
         im_m
         (fma
          (fma (* im_m im_m) 0.002777777777777778 0.08333333333333333)
          (* im_m (* im_m im_m))
          im_m)
         2.0))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double t_1 = t_0 * (exp(im_m) + exp(-im_m));
	double tmp;
	if (t_1 <= -5e+290) {
		tmp = cosh(im_m) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else if (t_1 <= 5e+22) {
		tmp = t_0 * fma(im_m, im_m, 2.0);
	} else {
		tmp = (0.5 * re) * fma(im_m, fma(fma((im_m * im_m), 0.002777777777777778, 0.08333333333333333), (im_m * (im_m * im_m)), im_m), 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(t_0 * Float64(exp(im_m) + exp(Float64(-im_m))))
	tmp = 0.0
	if (t_1 <= -5e+290)
		tmp = Float64(cosh(im_m) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	elseif (t_1 <= 5e+22)
		tmp = Float64(t_0 * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * fma(im_m, fma(fma(Float64(im_m * im_m), 0.002777777777777778, 0.08333333333333333), Float64(im_m * Float64(im_m * im_m)), im_m), 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+290], N[(N[Cosh[im$95$m], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+22], N[(t$95$0 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -4.9999999999999998e290

   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. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub0-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. neg-sub0 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. associate-*r/ N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.6

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

   9. Applied rewrites 74.6%

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

   if -4.9999999999999998e290 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.9999999999999996e22

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 4.9999999999999996e22 < (*.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 99.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.9

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

   8. Applied rewrites 64.9%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq -5 \cdot 10^{+290}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))