math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 28.8s

Alternatives: 25

Speedup: 1.5×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub0-neg N/A

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

   15. lower-neg.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-*l* N/A

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

   20. lower-*.f64 N/A

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

   21. lower-*.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 87.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot 0.5\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{t\_0}{\frac{1}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) 0.5)) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (* (cosh im) (fma re (* (* re re) -0.16666666666666666) re))
     (if (<= t_1 5e+132)
       (/
        t_0
        (/
         1.0
         (fma
          im
          (fma
           (* im im)
           (* im (fma (* im im) 0.002777777777777778 0.08333333333333333))
           im)
          2.0)))
       (*
        (cosh im)
        (fma
         (fma (* re re) 0.008333333333333333 -0.16666666666666666)
         (* re (* re re))
         re))))))

C

double code(double re, double im) {
	double t_0 = sin(re) * 0.5;
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(im) * fma(re, ((re * re) * -0.16666666666666666), re);
	} else if (t_1 <= 5e+132) {
		tmp = t_0 / (1.0 / fma(im, fma((im * im), (im * fma((im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0));
	} else {
		tmp = cosh(im) * fma(fma((re * re), 0.008333333333333333, -0.16666666666666666), (re * (re * re)), re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * 0.5)
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(im) * fma(re, Float64(Float64(re * re) * -0.16666666666666666), re));
	elseif (t_1 <= 5e+132)
		tmp = Float64(t_0 / Float64(1.0 / fma(im, fma(Float64(im * im), Float64(im * fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333)), im), 2.0)));
	else
		tmp = Float64(cosh(im) * fma(fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666), Float64(re * Float64(re * re)), re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[im], $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+132], N[(t$95$0 / N[(1.0 / N[(im * N[(N[(im * im), $MachinePrecision] * N[(im * N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 100.0

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

      6. lift-*.f64 N/A

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

      7. *-lft-identity 100.0

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

   6. Applied egg-rr 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 75.7

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

   9. Simplified 75.7%

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

   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))) < 5.0000000000000001e132

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   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. Simplified 99.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot \left(im \cdot im\right), im\right), 2\right)} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. flip-+ N/A

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

      9. clear-num N/A

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

   7. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{\sin re \cdot 0.5}{\frac{1}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)}}} \]

   if 5.0000000000000001e132 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied egg-rr 100.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 100.0

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

      6. lift-*.f64 N/A

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

      7. *-lft-identity 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 76.1

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

   9. Simplified 76.1%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re, \left(re \cdot re\right) \cdot -0.16666666666666666, re\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\sin re \cdot 0.5}{\frac{1}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, im \cdot \mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im\right), 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))