math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 16.5s

Alternatives: 22

Speedup: 1.0×

• 24.7% 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

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

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(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. Final simplification 100.0%

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

Alternative 2: 93.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re \cdot re, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (* t_1 (fma (* im im) -0.16666666666666666 1.0))
     (if (<= t_0 -0.04)
       (* (fma (fma 0.16666666666666666 re 0.5) (* re re) (+ 1.0 re)) (sin im))
       (if (<= t_0 1e-7)
         t_1
         (if (<= t_0 1.0)
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (sin im))
           t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * fma((im * im), -0.16666666666666666, 1.0);
	} else if (t_0 <= -0.04) {
		tmp = fma(fma(0.16666666666666666, re, 0.5), (re * re), (1.0 + re)) * sin(im);
	} else if (t_0 <= 1e-7) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(Float64(im * im), -0.16666666666666666, 1.0));
	elseif (t_0 <= -0.04)
		tmp = Float64(fma(fma(0.16666666666666666, re, 0.5), Float64(re * re), Float64(1.0 + re)) * sin(im));
	elseif (t_0 <= 1e-7)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(1.0 + re), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

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. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 75.9

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 75.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), \color{blue}{re \cdot re}, 1 + re\right) \cdot \sin im \]
   8. Step-by-step derivation

   9. Applied rewrites 75.9%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. distribute-lft1-in N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. lower-exp.f64 75.0

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

   12. Applied rewrites 75.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999995e-8 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

   if 9.9999999999999995e-8 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot \sin im \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(im \cdot e^{re}\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re \cdot re, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-7}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_1 \leq -0.04:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1 (* (sin im) (exp re)))
        (t_2
         (*
          (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
          (sin im))))
   (if (<= t_1 (- INFINITY))
     (* t_0 (fma (* im im) -0.16666666666666666 1.0))
     (if (<= t_1 -0.04)
       t_2
       (if (<= t_1 1e-7) t_0 (if (<= t_1 1.0) t_2 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = sin(im) * exp(re);
	double t_2 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 * fma((im * im), -0.16666666666666666, 1.0);
	} else if (t_1 <= -0.04) {
		tmp = t_2;
	} else if (t_1 <= 1e-7) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = Float64(sin(im) * exp(re))
	t_2 = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 * fma(Float64(im * im), -0.16666666666666666, 1.0));
	elseif (t_1 <= -0.04)
		tmp = t_2;
	elseif (t_1 <= 1e-7)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.04], t$95$2, If[LessEqual[t$95$1, 1e-7], t$95$0, If[LessEqual[t$95$1, 1.0], t$95$2, t$95$0]]]]]]]

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. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 75.9

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 75.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), \color{blue}{re \cdot re}, 1 + re\right) \cdot \sin im \]
   8. Step-by-step derivation

   9. Applied rewrites 75.9%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. distribute-lft1-in N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. lower-exp.f64 75.0

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

   12. Applied rewrites 75.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008 or 9.9999999999999995e-8 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999995e-8 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(im \cdot e^{re}\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-7}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (* t_1 (fma (* im im) -0.16666666666666666 1.0))
     (if (<= t_0 -0.04)
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       (if (<= t_0 1e-7)
         t_1
         (if (<= t_0 1.0) (* (+ (fma (* re re) 0.5 1.0) re) (sin im)) t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * fma((im * im), -0.16666666666666666, 1.0);
	} else if (t_0 <= -0.04) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else if (t_0 <= 1e-7) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = (fma((re * re), 0.5, 1.0) + re) * sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(Float64(im * im), -0.16666666666666666, 1.0));
	elseif (t_0 <= -0.04)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	elseif (t_0 <= 1e-7)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(Float64(fma(Float64(re * re), 0.5, 1.0) + re) * sin(im));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

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. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 75.9

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 75.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), \color{blue}{re \cdot re}, 1 + re\right) \cdot \sin im \]
   8. Step-by-step derivation

   9. Applied rewrites 75.9%

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

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. distribute-lft1-in N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. lower-exp.f64 75.0

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

   12. Applied rewrites 75.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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 97.1

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

   5. Applied rewrites 97.1%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999995e-8 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

   if 9.9999999999999995e-8 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(im \cdot e^{re}\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-7}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (* (fma (pow im 3.0) -0.16666666666666666 im) (+ 1.0 re))
     (if (<= t_0 -0.04)
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       (if (<= t_0 1e-7)
         t_1
         (if (<= t_0 1.0) (* (+ (fma (* re re) 0.5 1.0) re) (sin im)) t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(pow(im, 3.0), -0.16666666666666666, im) * (1.0 + re);
	} else if (t_0 <= -0.04) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else if (t_0 <= 1e-7) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = (fma((re * re), 0.5, 1.0) + re) * sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma((im ^ 3.0), -0.16666666666666666, im) * Float64(1.0 + re));
	elseif (t_0 <= -0.04)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	elseif (t_0 <= 1e-7)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(Float64(fma(Float64(re * re), 0.5, 1.0) + re) * sin(im));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision] * N[(1.0 + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

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. Taylor expanded in re around 0

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

      1. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 26.8

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 26.8%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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 97.1

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

   5. Applied rewrites 97.1%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999995e-8 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

   if 9.9999999999999995e-8 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-7}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ t_2 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re)))
        (t_1 (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)))
        (t_2 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (* (fma (pow im 3.0) -0.16666666666666666 im) (+ 1.0 re))
     (if (<= t_0 -0.04)
       t_1
       (if (<= t_0 1e-7) t_2 (if (<= t_0 1.0) t_1 t_2))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	double t_2 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(pow(im, 3.0), -0.16666666666666666, im) * (1.0 + re);
	} else if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= 1e-7) {
		tmp = t_2;
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im))
	t_2 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma((im ^ 3.0), -0.16666666666666666, im) * Float64(1.0 + re));
	elseif (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= 1e-7)
		tmp = t_2;
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision] * N[(1.0 + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, 1e-7], t$95$2, If[LessEqual[t$95$0, 1.0], t$95$1, t$95$2]]]]]]]

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. Taylor expanded in re around 0

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

      1. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 26.8

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 26.8%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008 or 9.9999999999999995e-8 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999995e-8 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-7}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := \left(1 + re\right) \cdot \sin im\\ t_2 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re)))
        (t_1 (* (+ 1.0 re) (sin im)))
        (t_2 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (* (fma (pow im 3.0) -0.16666666666666666 im) (+ 1.0 re))
     (if (<= t_0 -0.04)
       t_1
       (if (<= t_0 1e-7) t_2 (if (<= t_0 1.0) t_1 t_2))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = (1.0 + re) * sin(im);
	double t_2 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(pow(im, 3.0), -0.16666666666666666, im) * (1.0 + re);
	} else if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= 1e-7) {
		tmp = t_2;
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(Float64(1.0 + re) * sin(im))
	t_2 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma((im ^ 3.0), -0.16666666666666666, im) * Float64(1.0 + re));
	elseif (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= 1e-7)
		tmp = t_2;
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision] * N[(1.0 + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, 1e-7], t$95$2, If[LessEqual[t$95$0, 1.0], t$95$1, t$95$2]]]]]]]

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. Taylor expanded in re around 0

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

      1. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 26.8

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 26.8%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008 or 9.9999999999999995e-8 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999995e-8 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-7}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + re\right) \cdot \sin im\\ t_1 := \sin im \cdot e^{re}\\ t_2 := im \cdot e^{re}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_1 \leq -0.04:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 re) (sin im)))
        (t_1 (* (sin im) (exp re)))
        (t_2 (* im (exp re))))
   (if (<= t_1 (- INFINITY))
     (fma (pow im 3.0) -0.16666666666666666 im)
     (if (<= t_1 -0.04)
       t_0
       (if (<= t_1 1e-7) t_2 (if (<= t_1 1.0) t_0 t_2))))))

C

double code(double re, double im) {
	double t_0 = (1.0 + re) * sin(im);
	double t_1 = sin(im) * exp(re);
	double t_2 = im * exp(re);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(pow(im, 3.0), -0.16666666666666666, im);
	} else if (t_1 <= -0.04) {
		tmp = t_0;
	} else if (t_1 <= 1e-7) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(1.0 + re) * sin(im))
	t_1 = Float64(sin(im) * exp(re))
	t_2 = Float64(im * exp(re))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma((im ^ 3.0), -0.16666666666666666, im);
	elseif (t_1 <= -0.04)
		tmp = t_0;
	elseif (t_1 <= 1e-7)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision], If[LessEqual[t$95$1, -0.04], t$95$0, If[LessEqual[t$95$1, 1e-7], t$95$2, If[LessEqual[t$95$1, 1.0], t$95$0, t$95$2]]]]]]]

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. Taylor expanded in re around 0

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

      1. lower-sin.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 26.1%

      \[\leadsto \mathsf{fma}\left({im}^{3}, \color{blue}{-0.16666666666666666}, im\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008 or 9.9999999999999995e-8 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999995e-8 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-7}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (fma (pow im 3.0) -0.16666666666666666 im)
     (if (<= t_0 -0.04)
       (sin im)
       (if (<= t_0 1e-7) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(pow(im, 3.0), -0.16666666666666666, im);
	} else if (t_0 <= -0.04) {
		tmp = sin(im);
	} else if (t_0 <= 1e-7) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma((im ^ 3.0), -0.16666666666666666, im);
	elseif (t_0 <= -0.04)
		tmp = sin(im);
	elseif (t_0 <= 1e-7)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision], If[LessEqual[t$95$0, -0.04], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-7], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

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. Taylor expanded in re around 0

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

      1. lower-sin.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 26.1%

      \[\leadsto \mathsf{fma}\left({im}^{3}, \color{blue}{-0.16666666666666666}, im\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008 or 9.9999999999999995e-8 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   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 95.7

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

   5. Applied rewrites 95.7%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999995e-8 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.04:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-7}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.013888888888888888, re, -0.041666666666666664\right), re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right) \cdot im, re, im\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 (- INFINITY))
     (fma
      (*
       (fma
        (/
         (fma (fma -0.013888888888888888 re -0.041666666666666664) re 0.125)
         (*
          (fma re 0.16666666666666666 -0.5)
          (fma re 0.16666666666666666 -0.5)))
        re
        1.0)
       im)
      re
      im)
     (if (<= t_0 1.0)
       (sin im)
       (* (* (fma (fma 0.16666666666666666 re 0.5) re 1.0) re) im)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((fma((fma(fma(-0.013888888888888888, re, -0.041666666666666664), re, 0.125) / (fma(re, 0.16666666666666666, -0.5) * fma(re, 0.16666666666666666, -0.5))), re, 1.0) * im), re, im);
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = (fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(fma(Float64(fma(fma(-0.013888888888888888, re, -0.041666666666666664), re, 0.125) / Float64(fma(re, 0.16666666666666666, -0.5) * fma(re, 0.16666666666666666, -0.5))), re, 1.0) * im), re, im);
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(Float64(fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(-0.013888888888888888 * re + -0.041666666666666664), $MachinePrecision] * re + 0.125), $MachinePrecision] / N[(N[(re * 0.16666666666666666 + -0.5), $MachinePrecision] * N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * 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. 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 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.6%

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

   10. Applied rewrites 9.0%

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\left(0.027777777777777776 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) - \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot 0.25}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right), re, im\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\frac{1}{8} + re \cdot \left(\frac{-1}{72} \cdot re - \frac{1}{24}\right)}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right) \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right)}, re, 1\right), re, im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 3.1%

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.013888888888888888, re, -0.041666666666666664\right), re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right), re, im\right) \]

   if -inf.0 < (*.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 62.5

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

   5. Applied rewrites 62.5%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 30.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 30.6%

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

   13. Applied rewrites 36.6%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.013888888888888888, re, -0.041666666666666664\right), re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right) \cdot im, re, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.013888888888888888, re, -0.041666666666666664\right), re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right) \cdot im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right) \cdot \mathsf{fma}\left(0.16666666666666666, re, -0.5\right)}{-\mathsf{fma}\left(0.16666666666666666, re, -0.5\right)} \cdot \frac{-1}{\mathsf{fma}\left(0.16666666666666666, re, -0.5\right)}, re, 1\right) \cdot im, re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) -0.15)
   (fma
    (*
     (fma
      (/
       (fma (fma -0.013888888888888888 re -0.041666666666666664) re 0.125)
       (* (fma re 0.16666666666666666 -0.5) (fma re 0.16666666666666666 -0.5)))
      re
      1.0)
     im)
    re
    im)
   (fma
    (*
     (fma
      (*
       (/
        (*
         (fma (* re re) 0.027777777777777776 -0.25)
         (fma 0.16666666666666666 re -0.5))
        (- (fma 0.16666666666666666 re -0.5)))
       (/ -1.0 (fma 0.16666666666666666 re -0.5)))
      re
      1.0)
     im)
    re
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= -0.15) {
		tmp = fma((fma((fma(fma(-0.013888888888888888, re, -0.041666666666666664), re, 0.125) / (fma(re, 0.16666666666666666, -0.5) * fma(re, 0.16666666666666666, -0.5))), re, 1.0) * im), re, im);
	} else {
		tmp = fma((fma((((fma((re * re), 0.027777777777777776, -0.25) * fma(0.16666666666666666, re, -0.5)) / -fma(0.16666666666666666, re, -0.5)) * (-1.0 / fma(0.16666666666666666, re, -0.5))), re, 1.0) * im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= -0.15)
		tmp = fma(Float64(fma(Float64(fma(fma(-0.013888888888888888, re, -0.041666666666666664), re, 0.125) / Float64(fma(re, 0.16666666666666666, -0.5) * fma(re, 0.16666666666666666, -0.5))), re, 1.0) * im), re, im);
	else
		tmp = fma(Float64(fma(Float64(Float64(Float64(fma(Float64(re * re), 0.027777777777777776, -0.25) * fma(0.16666666666666666, re, -0.5)) / Float64(-fma(0.16666666666666666, re, -0.5))) * Float64(-1.0 / fma(0.16666666666666666, re, -0.5))), re, 1.0) * im), re, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], -0.15], N[(N[(N[(N[(N[(N[(-0.013888888888888888 * re + -0.041666666666666664), $MachinePrecision] * re + 0.125), $MachinePrecision] / N[(N[(re * 0.16666666666666666 + -0.5), $MachinePrecision] * N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.027777777777777776 + -0.25), $MachinePrecision] * N[(0.16666666666666666 * re + -0.5), $MachinePrecision]), $MachinePrecision] / (-N[(0.16666666666666666 * re + -0.5), $MachinePrecision])), $MachinePrecision] * N[(-1.0 / N[(0.16666666666666666 * re + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.149999999999999994

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 45.4

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.7%

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

   10. Applied rewrites 6.4%

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\left(0.027777777777777776 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) - \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot 0.25}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right), re, im\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\frac{1}{8} + re \cdot \left(\frac{-1}{72} \cdot re - \frac{1}{24}\right)}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right) \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right)}, re, 1\right), re, im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 2.9%

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.013888888888888888, re, -0.041666666666666664\right), re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right), re, im\right) \]

   if -0.149999999999999994 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 39.8%

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

   10. Applied rewrites 36.3%

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\left(0.027777777777777776 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) - \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot 0.25}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right), re, im\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 40.8%

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(0.16666666666666666, re, -0.5\right)} \cdot \frac{\mathsf{fma}\left(0.16666666666666666, re, -0.5\right) \cdot \mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right)}{-\mathsf{fma}\left(0.16666666666666666, re, -0.5\right)}, re, 1\right), re, im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.013888888888888888, re, -0.041666666666666664\right), re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right) \cdot im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(re \cdot re, 0.027777777777777776, -0.25\right) \cdot \mathsf{fma}\left(0.16666666666666666, re, -0.5\right)}{-\mathsf{fma}\left(0.16666666666666666, re, -0.5\right)} \cdot \frac{-1}{\mathsf{fma}\left(0.16666666666666666, re, -0.5\right)}, re, 1\right) \cdot im, re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.013888888888888888, re, -0.041666666666666664\right), re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right) \cdot im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) -0.15)
   (fma
    (*
     (fma
      (/
       (fma (fma -0.013888888888888888 re -0.041666666666666664) re 0.125)
       (* (fma re 0.16666666666666666 -0.5) (fma re 0.16666666666666666 -0.5)))
      re
      1.0)
     im)
    re
    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 ((sin(im) * exp(re)) <= -0.15) {
		tmp = fma((fma((fma(fma(-0.013888888888888888, re, -0.041666666666666664), re, 0.125) / (fma(re, 0.16666666666666666, -0.5) * fma(re, 0.16666666666666666, -0.5))), re, 1.0) * im), re, 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(sin(im) * exp(re)) <= -0.15)
		tmp = fma(Float64(fma(Float64(fma(fma(-0.013888888888888888, re, -0.041666666666666664), re, 0.125) / Float64(fma(re, 0.16666666666666666, -0.5) * fma(re, 0.16666666666666666, -0.5))), re, 1.0) * im), re, 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[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], -0.15], N[(N[(N[(N[(N[(N[(-0.013888888888888888 * re + -0.041666666666666664), $MachinePrecision] * re + 0.125), $MachinePrecision] / N[(N[(re * 0.16666666666666666 + -0.5), $MachinePrecision] * N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision] * re + 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)) < -0.149999999999999994

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 45.4

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.7%

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

   10. Applied rewrites 6.4%

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\left(0.027777777777777776 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) - \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot 0.25}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right), re, im\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\frac{1}{8} + re \cdot \left(\frac{-1}{72} \cdot re - \frac{1}{24}\right)}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right) \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right)}, re, 1\right), re, im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 2.9%

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.013888888888888888, re, -0.041666666666666664\right), re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right), re, im\right) \]

   if -0.149999999999999994 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.8%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.013888888888888888, re, -0.041666666666666664\right), re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right) \cdot im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.041666666666666664, re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right) \cdot im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.97)
   (fma
    (*
     (fma
      (/
       (fma -0.041666666666666664 re 0.125)
       (* (fma re 0.16666666666666666 -0.5) (fma re 0.16666666666666666 -0.5)))
      re
      1.0)
     im)
    re
    im)
   (* (* (fma (fma 0.16666666666666666 re 0.5) re 1.0) re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.97) {
		tmp = fma((fma((fma(-0.041666666666666664, re, 0.125) / (fma(re, 0.16666666666666666, -0.5) * fma(re, 0.16666666666666666, -0.5))), re, 1.0) * im), re, im);
	} else {
		tmp = (fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.97)
		tmp = fma(Float64(fma(Float64(fma(-0.041666666666666664, re, 0.125) / Float64(fma(re, 0.16666666666666666, -0.5) * fma(re, 0.16666666666666666, -0.5))), re, 1.0) * im), re, im);
	else
		tmp = Float64(Float64(fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.97], N[(N[(N[(N[(N[(-0.041666666666666664 * re + 0.125), $MachinePrecision] / N[(N[(re * 0.16666666666666666 + -0.5), $MachinePrecision] * N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re), $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. 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 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.4%

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

   10. Applied rewrites 32.7%

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\left(0.027777777777777776 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) - \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot 0.25}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right), re, im\right) \]

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\frac{1}{8} + \frac{-1}{24} \cdot re}{\mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right) \cdot \mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right)}, re, 1\right), re, im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 34.3%

       \[\leadsto \mathsf{fma}\left(im \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.041666666666666664, re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right), re, im\right) \]

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

   1. Initial program 100.0%

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

   3. 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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.8%

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

   13. Applied rewrites 30.7%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.97:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.041666666666666664, re, 0.125\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)}, re, 1\right) \cdot im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.97:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.97)
   (* 1.0 im)
   (* (* (fma 0.5 re 1.0) im) re)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.97) {
		tmp = 1.0 * im;
	} else {
		tmp = (fma(0.5, re, 1.0) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.97)
		tmp = Float64(1.0 * im);
	else
		tmp = Float64(Float64(fma(0.5, re, 1.0) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.97], N[(1.0 * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * im), $MachinePrecision] * re), $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. 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 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto 1 \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 31.9%

      \[\leadsto 1 \cdot im \]

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

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.8%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 15.5%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.97:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 92.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ \mathbf{if}\;re \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{-20}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re))))
   (if (<= re -2.1e-5) t_0 (if (<= re 3.5e-20) (sin im) t_0))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double tmp;
	if (re <= -2.1e-5) {
		tmp = t_0;
	} else if (re <= 3.5e-20) {
		tmp = sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * exp(re)
    if (re <= (-2.1d-5)) then
        tmp = t_0
    else if (re <= 3.5d-20) then
        tmp = sin(im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * Math.exp(re);
	double tmp;
	if (re <= -2.1e-5) {
		tmp = t_0;
	} else if (re <= 3.5e-20) {
		tmp = Math.sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * math.exp(re)
	tmp = 0
	if re <= -2.1e-5:
		tmp = t_0
	elif re <= 3.5e-20:
		tmp = math.sin(im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	tmp = 0.0
	if (re <= -2.1e-5)
		tmp = t_0;
	elseif (re <= 3.5e-20)
		tmp = sin(im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * exp(re);
	tmp = 0.0;
	if (re <= -2.1e-5)
		tmp = t_0;
	elseif (re <= 3.5e-20)
		tmp = sin(im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -2.1e-5], t$95$0, If[LessEqual[re, 3.5e-20], N[Sin[im], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -2.09999999999999988e-5 or 3.50000000000000003e-20 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 84.5

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

   5. Applied rewrites 84.5%

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

   if -2.09999999999999988e-5 < re < 3.50000000000000003e-20

   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 98.5

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

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\sin im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{-20}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im))

C

double code(double re, double im) {
	return fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
}

Julia

function code(re, im)
	return Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im)
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
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 71.7

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

5. Applied rewrites 71.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 39.2%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im \]
9. Add Preprocessing

Alternative 17: 38.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot im, re, im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (fma (* (fma (fma 0.16666666666666666 re 0.5) re 1.0) im) re im))

C

double code(double re, double im) {
	return fma((fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * im), re, im);
}

Julia

function code(re, im)
	return fma(Float64(fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * im), re, im)
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
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 71.7

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

5. Applied rewrites 71.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 38.1%

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

9. Final simplification 38.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot im, re, im\right) \]
10. Add Preprocessing

Alternative 18: 37.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (fma (fma 0.5 re 1.0) re 1.0) im))

C

double code(double re, double im) {
	return fma(fma(0.5, re, 1.0), re, 1.0) * im;
}

Julia

function code(re, im)
	return Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im)
end

Wolfram

code[re_, im_] := N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
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 71.7

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

5. Applied rewrites 71.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 37.9%

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

Alternative 19: 34.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot im, re, im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (fma (* (fma 0.5 re 1.0) im) re im))

C

double code(double re, double im) {
	return fma((fma(0.5, re, 1.0) * im), re, im);
}

Julia

function code(re, im)
	return fma(Float64(fma(0.5, re, 1.0) * im), re, im)
end

Wolfram

code[re_, im_] := N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
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 71.7

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

5. Applied rewrites 71.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 38.1%

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

9. Taylor expanded in re around 0

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

11. Applied rewrites 35.1%

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

12. Final simplification 35.1%

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

Alternative 20: 28.0% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 10^{+68}:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1e+68) (* 1.0 im) (* im re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1e+68) {
		tmp = 1.0 * im;
	} else {
		tmp = im * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1d+68) then
        tmp = 1.0d0 * im
    else
        tmp = im * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1e+68) {
		tmp = 1.0 * im;
	} else {
		tmp = im * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1e+68:
		tmp = 1.0 * im
	else:
		tmp = im * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1e+68)
		tmp = Float64(1.0 * im);
	else
		tmp = Float64(im * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1e+68)
		tmp = 1.0 * im;
	else
		tmp = im * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1e+68], N[(1.0 * im), $MachinePrecision], N[(im * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.99999999999999953e67

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   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 80.3

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

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto 1 \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 34.1%

      \[\leadsto 1 \cdot im \]

   if 9.99999999999999953e67 < 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 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 9.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 9.9%

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

   12. Taylor expanded in re around 0

       \[\leadsto im \cdot re \]
   13. Step-by-step derivation

   14. Applied rewrites 6.4%

       \[\leadsto im \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 29.6% 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. 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 71.7

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

5. Applied rewrites 71.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 30.1%

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

Alternative 22: 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. 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 71.7

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

5. Applied rewrites 71.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 38.1%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 14.0%

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

12. Taylor expanded in re around 0

    \[\leadsto im \cdot re \]
13. Step-by-step derivation

14. Applied rewrites 6.3%

    \[\leadsto im \cdot re \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))