math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 17

Speedup: 1.0×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin im}{e^{-re}} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (sin im) (exp (- re))))

C

double code(double re, double im) {
	return sin(im) / exp(-re);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(im) / exp(-re)
end function

Java

public static double code(double re, double im) {
	return Math.sin(im) / Math.exp(-re);
}

Python

def code(re, im):
	return math.sin(im) / math.exp(-re)

Julia

function code(re, im)
	return Float64(sin(im) / exp(Float64(-re)))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(im) / exp(-re);
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh---cosh-rev N/A

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

   6. associate-*l/ N/A

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

   7. sinh-cosh N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 100.0

       \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]
5. Add Preprocessing

Alternative 2: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(t\_1 \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\left(1.5 - \mathsf{fma}\left(-0.5, re, 0.5\right)\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-31} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (* (* (* t_1 im) -0.16666666666666666) im)
     (if (<= t_0 -0.05)
       (* (- 1.5 (fma -0.5 re 0.5)) (sin im))
       (if (or (<= t_0 1e-31) (not (<= t_0 1.0)))
         t_1
         (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((t_1 * im) * -0.16666666666666666) * im;
	} else if (t_0 <= -0.05) {
		tmp = (1.5 - fma(-0.5, re, 0.5)) * sin(im);
	} else if ((t_0 <= 1e-31) || !(t_0 <= 1.0)) {
		tmp = t_1;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_1 * im) * -0.16666666666666666) * im);
	elseif (t_0 <= -0.05)
		tmp = Float64(Float64(1.5 - fma(-0.5, re, 0.5)) * sin(im));
	elseif ((t_0 <= 1e-31) || !(t_0 <= 1.0))
		tmp = t_1;
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(t$95$1 * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[(1.5 - N[(-0.5 * re + 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-31], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], t$95$1, N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in im around inf

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

   10. Applied rewrites 16.7%

       \[\leadsto \left(\left(\left(e^{re} \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right) \cdot im \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 99.0%

      \[\leadsto \left(\color{blue}{1.5} - \frac{e^{-re}}{2}\right) \cdot \sin im \]

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.0

         \[\leadsto \left(1.5 - \color{blue}{\mathsf{fma}\left(-0.5, re, 0.5\right)}\right) \cdot \sin im \]

   10. Applied rewrites 99.0%

       \[\leadsto \left(1.5 - \color{blue}{\mathsf{fma}\left(-0.5, re, 0.5\right)}\right) \cdot \sin im \]

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-31 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      3. lower-exp.f64 96.1

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   5. Applied rewrites 96.1%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   if 1e-31 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\left(\left(e^{re} \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\left(1.5 - \mathsf{fma}\left(-0.5, re, 0.5\right)\right) \cdot \sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-31} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot -0.16666666666666666\right) \cdot im\right) \cdot im\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\left(1.5 - \mathsf{fma}\left(-0.5, re, 0.5\right)\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-31} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (*
        (*
         (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
         -0.16666666666666666)
        im)
       im)
      im)
     (if (<= t_0 -0.05)
       (* (- 1.5 (fma -0.5 re 0.5)) (sin im))
       (if (or (<= t_0 1e-31) (not (<= t_0 1.0)))
         (* (exp re) im)
         (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (((fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * -0.16666666666666666) * im) * im) * im;
	} else if (t_0 <= -0.05) {
		tmp = (1.5 - fma(-0.5, re, 0.5)) * sin(im);
	} else if ((t_0 <= 1e-31) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * -0.16666666666666666) * im) * im) * im);
	elseif (t_0 <= -0.05)
		tmp = Float64(Float64(1.5 - fma(-0.5, re, 0.5)) * sin(im));
	elseif ((t_0 <= 1e-31) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[(1.5 - N[(-0.5 * re + 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-31], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 37.2%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 13.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 99.0%

      \[\leadsto \left(\color{blue}{1.5} - \frac{e^{-re}}{2}\right) \cdot \sin im \]

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.0

         \[\leadsto \left(1.5 - \color{blue}{\mathsf{fma}\left(-0.5, re, 0.5\right)}\right) \cdot \sin im \]

   10. Applied rewrites 99.0%

       \[\leadsto \left(1.5 - \color{blue}{\mathsf{fma}\left(-0.5, re, 0.5\right)}\right) \cdot \sin im \]

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-31 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      3. lower-exp.f64 96.1

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   5. Applied rewrites 96.1%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   if 1e-31 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot -0.16666666666666666\right) \cdot im\right) \cdot im\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\left(1.5 - \mathsf{fma}\left(-0.5, re, 0.5\right)\right) \cdot \sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-31} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot -0.16666666666666666\right) \cdot im\right) \cdot im\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\left(1.5 - \mathsf{fma}\left(-0.5, re, 0.5\right)\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-31} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (*
        (*
         (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
         -0.16666666666666666)
        im)
       im)
      im)
     (if (<= t_0 -0.05)
       (* (- 1.5 (fma -0.5 re 0.5)) (sin im))
       (if (or (<= t_0 1e-31) (not (<= t_0 1.0)))
         (* (exp re) im)
         (* (+ 1.0 re) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (((fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * -0.16666666666666666) * im) * im) * im;
	} else if (t_0 <= -0.05) {
		tmp = (1.5 - fma(-0.5, re, 0.5)) * sin(im);
	} else if ((t_0 <= 1e-31) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = (1.0 + re) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * -0.16666666666666666) * im) * im) * im);
	elseif (t_0 <= -0.05)
		tmp = Float64(Float64(1.5 - fma(-0.5, re, 0.5)) * sin(im));
	elseif ((t_0 <= 1e-31) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(Float64(1.0 + re) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[(1.5 - N[(-0.5 * re + 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-31], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 37.2%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 13.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 99.0%

      \[\leadsto \left(\color{blue}{1.5} - \frac{e^{-re}}{2}\right) \cdot \sin im \]

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.0

         \[\leadsto \left(1.5 - \color{blue}{\mathsf{fma}\left(-0.5, re, 0.5\right)}\right) \cdot \sin im \]

   10. Applied rewrites 99.0%

       \[\leadsto \left(1.5 - \color{blue}{\mathsf{fma}\left(-0.5, re, 0.5\right)}\right) \cdot \sin im \]

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-31 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      3. lower-exp.f64 96.1

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   5. Applied rewrites 96.1%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   if 1e-31 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \sin im \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot -0.16666666666666666\right) \cdot im\right) \cdot im\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\left(1.5 - \mathsf{fma}\left(-0.5, re, 0.5\right)\right) \cdot \sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-31} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot -0.16666666666666666\right) \cdot im\right) \cdot im\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-31} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (*
        (*
         (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
         -0.16666666666666666)
        im)
       im)
      im)
     (if (<= t_0 -0.05)
       (sin im)
       (if (or (<= t_0 1e-31) (not (<= t_0 1.0)))
         (* (exp re) im)
         (* (+ 1.0 re) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (((fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * -0.16666666666666666) * im) * im) * im;
	} else if (t_0 <= -0.05) {
		tmp = sin(im);
	} else if ((t_0 <= 1e-31) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = (1.0 + re) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * -0.16666666666666666) * im) * im) * im);
	elseif (t_0 <= -0.05)
		tmp = sin(im);
	elseif ((t_0 <= 1e-31) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(Float64(1.0 + re) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Sin[im], $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-31], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 37.2%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 13.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-31 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      3. lower-exp.f64 96.1

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   5. Applied rewrites 96.1%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   if 1e-31 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \sin im \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot -0.16666666666666666\right) \cdot im\right) \cdot im\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-31} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot -0.16666666666666666\right) \cdot im\right) \cdot im\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq -0.05 \lor \neg \left(t\_0 \leq 10^{-31} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (*
        (*
         (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
         -0.16666666666666666)
        im)
       im)
      im)
     (if (or (<= t_0 -0.05) (not (or (<= t_0 1e-31) (not (<= t_0 1.0)))))
       (sin im)
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (((fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * -0.16666666666666666) * im) * im) * im;
	} else if ((t_0 <= -0.05) || !((t_0 <= 1e-31) || !(t_0 <= 1.0))) {
		tmp = sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * -0.16666666666666666) * im) * im) * im);
	elseif ((t_0 <= -0.05) || !((t_0 <= 1e-31) || !(t_0 <= 1.0)))
		tmp = sin(im);
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.05], N[Not[Or[LessEqual[t$95$0, 1e-31], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[Sin[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 37.2%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 13.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1e-31 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-31 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot e^{re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      3. lower-exp.f64 96.1

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   5. Applied rewrites 96.1%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot -0.16666666666666666\right) \cdot im\right) \cdot im\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05 \lor \neg \left(e^{re} \cdot \sin im \leq 10^{-31} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im, re, im\right) \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (* (* (fma im re im) im) -0.16666666666666666) im)
   (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = ((fma(im, re, im) * im) * -0.16666666666666666) * im;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(im * re + im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 30.4%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 12.9%

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

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

          \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \cdot im \]

      4. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      6. lower-exp.f64 56.8

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   7. Applied rewrites 56.8%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 50.3%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im, re, im\right) \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 28.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im, re, im\right) \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, im\right), re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (* (* (fma im re im) im) -0.16666666666666666) im)
   (fma (fma (* im (fma 0.16666666666666666 re 0.5)) re im) re im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = ((fma(im, re, im) * im) * -0.16666666666666666) * im;
	} else {
		tmp = fma(fma((im * fma(0.16666666666666666, re, 0.5)), re, im), re, im);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(im * re + im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(im * N[(0.16666666666666666 * re + 0.5), $MachinePrecision]), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 30.4%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 12.9%

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

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

          \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \cdot im \]

      4. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      6. lower-exp.f64 56.8

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   7. Applied rewrites 56.8%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 48.5%

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

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im, re, im\right) \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, im\right), re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im, re, im\right) \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (* (* (fma im re im) im) -0.16666666666666666) im)
   (* (fma (fma 0.5 re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = ((fma(im, re, im) * im) * -0.16666666666666666) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(im * re + im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 30.4%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 12.9%

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

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

          \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \cdot im \]

      4. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      6. lower-exp.f64 56.8

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   7. Applied rewrites 56.8%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 47.3%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im, re, im\right) \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (fma (* im im) -0.16666666666666666 1.0) im)
   (* (fma (fma 0.5 re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = fma((im * im), -0.16666666666666666, 1.0) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 29.3%

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

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

          \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \cdot im \]

      4. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      6. lower-exp.f64 56.8

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   7. Applied rewrites 56.8%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 47.3%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 5e-7)
   (* (fma (* im im) -0.16666666666666666 1.0) im)
   (* (* (fma 0.5 re 1.0) re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 5e-7) {
		tmp = fma((im * im), -0.16666666666666666, 1.0) * im;
	} else {
		tmp = (fma(0.5, re, 1.0) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 5e-7)
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * im);
	else
		tmp = Float64(Float64(fma(0.5, re, 1.0) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999977e-7

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 72.3

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

   4. Applied rewrites 72.3%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 41.2%

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

   if 4.99999999999999977e-7 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 99.9

          \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \cdot im \]

      4. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      6. lower-exp.f64 36.0

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   7. Applied rewrites 36.0%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 22.0%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 22.3%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 5e-7)
   (* (fma (* im im) -0.16666666666666666 1.0) im)
   (* (* (* re re) 0.5) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 5e-7) {
		tmp = fma((im * im), -0.16666666666666666, 1.0) * im;
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 5e-7)
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * im);
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999977e-7

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 72.3

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

   4. Applied rewrites 72.3%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 41.2%

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

   if 4.99999999999999977e-7 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 99.9

          \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \cdot im \]

      4. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      6. lower-exp.f64 36.0

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   7. Applied rewrites 36.0%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 22.0%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 22.1%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.97) (fma re im im) (* (* (* re re) 0.5) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.97) {
		tmp = fma(re, im, im);
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.97)
		tmp = fma(re, im, im);
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.97], N[(re * im + im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.96999999999999997

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

          \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \cdot im \]

      4. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      6. lower-exp.f64 70.4

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   7. Applied rewrites 70.4%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   8. Taylor expanded in re around 0

      \[\leadsto im + \color{blue}{im \cdot re} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.2%

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

   if 0.96999999999999997 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

          \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \cdot im \]

      4. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      6. lower-exp.f64 62.9

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   7. Applied rewrites 62.9%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 37.2%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 37.4%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 15: 71.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -78:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im, re, im\right) \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\\ \mathbf{elif}\;re \leq 0.00061:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+93}:\\ \;\;\;\;\left(\left(1 + re\right) \cdot \left(\left(\left({\left(im \cdot im\right)}^{-1} - 0.16666666666666666\right) \cdot im\right) \cdot im\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -78.0)
   (* (* (* (fma im re im) im) -0.16666666666666666) im)
   (if (<= re 0.00061)
     (sin im)
     (if (<= re 1.25e+93)
       (*
        (*
         (+ 1.0 re)
         (* (* (- (pow (* im im) -1.0) 0.16666666666666666) im) im))
        im)
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -78.0) {
		tmp = ((fma(im, re, im) * im) * -0.16666666666666666) * im;
	} else if (re <= 0.00061) {
		tmp = sin(im);
	} else if (re <= 1.25e+93) {
		tmp = ((1.0 + re) * (((pow((im * im), -1.0) - 0.16666666666666666) * im) * im)) * im;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -78.0)
		tmp = Float64(Float64(Float64(fma(im, re, im) * im) * -0.16666666666666666) * im);
	elseif (re <= 0.00061)
		tmp = sin(im);
	elseif (re <= 1.25e+93)
		tmp = Float64(Float64(Float64(1.0 + re) * Float64(Float64(Float64((Float64(im * im) ^ -1.0) - 0.16666666666666666) * im) * im)) * im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -78.0], N[(N[(N[(N[(im * re + im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 0.00061], N[Sin[im], $MachinePrecision], If[LessEqual[re, 1.25e+93], N[(N[(N[(1.0 + re), $MachinePrecision] * N[(N[(N[(N[Power[N[(im * im), $MachinePrecision], -1.0], $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -78

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 2.5%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 25.0%

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

   if -78 < re < 6.09999999999999974e-4

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 6.09999999999999974e-4 < re < 1.25e93

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. sinh-def N/A

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

      4. lift-exp.f64 N/A

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

      5. div-sub N/A

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cosh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(e^{re}, 1, 0\right)\right) \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 25.2%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 43.8%

       \[\leadsto \left(\left(1 + re\right) \cdot \left(\left(\left(\frac{1}{im \cdot im} - 0.16666666666666666\right) \cdot im\right) \cdot im\right)\right) \cdot im \]

   if 1.25e93 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

          \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \cdot im \]

      4. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot im} \]

      6. lower-exp.f64 85.0

         \[\leadsto \color{blue}{e^{re}} \cdot im \]

   7. Applied rewrites 85.0%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 85.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -78:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im, re, im\right) \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\\ \mathbf{elif}\;re \leq 0.00061:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+93}:\\ \;\;\;\;\left(\left(1 + re\right) \cdot \left(\left(\left({\left(im \cdot im\right)}^{-1} - 0.16666666666666666\right) \cdot im\right) \cdot im\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.3% accurate, 29.4× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (fma re im im))

C

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

Julia

function code(re, im)
	return fma(re, im, im)
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh---cosh-rev N/A

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

   6. associate-*l/ N/A

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

   7. sinh-cosh N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 100.0

       \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
6. Step-by-step derivation

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. rec-exp N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \cdot im \]

   4. remove-double-div N/A

      \[\leadsto \color{blue}{e^{re}} \cdot im \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   6. lower-exp.f64 69.3

      \[\leadsto \color{blue}{e^{re}} \cdot im \]

7. Applied rewrites 69.3%

   \[\leadsto \color{blue}{e^{re} \cdot im} \]

8. Taylor expanded in re around 0

   \[\leadsto im + \color{blue}{im \cdot re} \]
9. Step-by-step derivation

10. Applied rewrites 35.9%

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

11. Final simplification 35.9%

    \[\leadsto \mathsf{fma}\left(re, im, im\right) \]
12. Add Preprocessing

Alternative 17: 6.8% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh---cosh-rev N/A

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

   6. associate-*l/ N/A

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

   7. sinh-cosh N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 100.0

       \[\leadsto \frac{\sin im}{e^{\color{blue}{-re}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
6. Step-by-step derivation

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. rec-exp N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \cdot im \]

   4. remove-double-div N/A

      \[\leadsto \color{blue}{e^{re}} \cdot im \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   6. lower-exp.f64 69.3

      \[\leadsto \color{blue}{e^{re}} \cdot im \]

7. Applied rewrites 69.3%

   \[\leadsto \color{blue}{e^{re} \cdot im} \]

8. Taylor expanded in re around 0

   \[\leadsto im + \color{blue}{im \cdot re} \]
9. Step-by-step derivation

10. Applied rewrites 35.9%

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

11. Taylor expanded in re around inf

    \[\leadsto im \cdot re \]
12. Step-by-step derivation

13. Applied rewrites 9.3%

    \[\leadsto im \cdot re \]

14. Final simplification 9.3%

    \[\leadsto im \cdot re \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))