math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 11.7s

Alternatives: 21

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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: 92.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right) + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -0.1)
       (sin im)
       (if (<= t_0 5e-46)
         t_1
         (if (<= t_0 10.0)
           (*
            (+ (fma (* re re) (fma 0.16666666666666666 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 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 5e-46) {
		tmp = t_1;
	} else if (t_0 <= 10.0) {
		tmp = (fma((re * re), fma(0.16666666666666666, 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 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 5e-46)
		tmp = t_1;
	elseif (t_0 <= 10.0)
		tmp = Float64(Float64(fma(Float64(re * re), fma(0.16666666666666666, 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, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 5e-46], t$95$1, If[LessEqual[t$95$0, 10.0], N[(N[(N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 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 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.8%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 54.4%

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

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   12. 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

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

       3. associate-*r* N/A

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

       4. distribute-rgt-in N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       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. lower-*.f64 N/A

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

       18. lower-exp.f64 75.0

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

   13. Applied rewrites 75.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   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 97.6

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

   5. Applied rewrites 97.6%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999992e-46 or 10 < (*.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 96.0

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

   5. Applied rewrites 96.0%

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

   if 4.99999999999999992e-46 < (*.f64 (exp.f64 re) (sin.f64 im)) < 10

   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 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(im \cdot e^{re}\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right) + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -0.1)
       (sin im)
       (if (<= t_0 5e-46)
         t_1
         (if (<= t_0 10.0)
           (*
            (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 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 5e-46) {
		tmp = t_1;
	} else if (t_0 <= 10.0) {
		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 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 5e-46)
		tmp = t_1;
	elseif (t_0 <= 10.0)
		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, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 5e-46], t$95$1, If[LessEqual[t$95$0, 10.0], 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 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.8%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 54.4%

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

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   12. 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

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

       3. associate-*r* N/A

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

       4. distribute-rgt-in N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       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. lower-*.f64 N/A

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

       18. lower-exp.f64 75.0

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

   13. Applied rewrites 75.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   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 97.6

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

   5. Applied rewrites 97.6%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999992e-46 or 10 < (*.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 96.0

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

   5. Applied rewrites 96.0%

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

   if 4.99999999999999992e-46 < (*.f64 (exp.f64 re) (sin.f64 im)) < 10

   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(im \cdot e^{re}\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-206}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -0.1)
       (sin im)
       (if (<= t_0 1e-206)
         t_1
         (if (<= t_0 10.0)
           (* (+ (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 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 1e-206) {
		tmp = t_1;
	} else if (t_0 <= 10.0) {
		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 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 1e-206)
		tmp = t_1;
	elseif (t_0 <= 10.0)
		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, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-206], t$95$1, If[LessEqual[t$95$0, 10.0], 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 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.8%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 54.4%

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

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{im \cdot \left(e^{re} + \frac{-1}{6} \cdot \left({im}^{2} \cdot e^{re}\right)\right)} \]
   12. 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

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

       3. associate-*r* N/A

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

       4. distribute-rgt-in N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       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. lower-*.f64 N/A

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

       18. lower-exp.f64 75.0

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

   13. Applied rewrites 75.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   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 97.6

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

   5. Applied rewrites 97.6%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.00000000000000003e-206 or 10 < (*.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 95.5

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

   5. Applied rewrites 95.5%

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

   if 1.00000000000000003e-206 < (*.f64 (exp.f64 re) (sin.f64 im)) < 10

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(im \cdot e^{re}\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-206}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{re}{im}, \mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 -0.1)
     (* (fma (* -0.16666666666666666 (* im im)) im im) (+ 1.0 re))
     (if (<= t_0 0.0)
       (/
        1.0
        (fma
         (fma (/ re im) (fma -0.16666666666666666 re 0.5) (/ -1.0 im))
         re
         (/ 1.0 im)))
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = fma((-0.16666666666666666 * (im * im)), im, im) * (1.0 + re);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / fma(fma((re / im), fma(-0.16666666666666666, re, 0.5), (-1.0 / im)), re, (1.0 / 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)
	t_0 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im) * Float64(1.0 + re));
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 / fma(fma(Float64(re / im), fma(-0.16666666666666666, re, 0.5), Float64(-1.0 / im)), re, Float64(1.0 / 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_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision] * N[(1.0 + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(N[(N[(re / im), $MachinePrecision] * N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] + N[(-1.0 / im), $MachinePrecision]), $MachinePrecision] * re + N[(1.0 / im), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   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 51.5

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

   5. Applied rewrites 51.5%

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

   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 9.3

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

   8. Applied rewrites 9.3%

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

   10. Applied rewrites 9.3%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0

   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 98.9

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

   5. Applied rewrites 98.9%

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

   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 44.2%

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

   10. Applied rewrites 44.1%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 76.7%

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

   if -0.0 < (*.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 65.8

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

   5. Applied rewrites 65.8%

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

   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 55.7%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{re}{im}, \mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{im}, re, \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 -0.1)
     (* (fma (* -0.16666666666666666 (* im im)) im im) (+ 1.0 re))
     (if (<= t_0 0.0)
       (/ 1.0 (fma (fma (/ 0.5 im) re (/ -1.0 im)) re (/ 1.0 im)))
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = fma((-0.16666666666666666 * (im * im)), im, im) * (1.0 + re);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / fma(fma((0.5 / im), re, (-1.0 / im)), re, (1.0 / 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)
	t_0 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im) * Float64(1.0 + re));
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 / fma(fma(Float64(0.5 / im), re, Float64(-1.0 / im)), re, Float64(1.0 / 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_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision] * N[(1.0 + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(N[(N[(0.5 / im), $MachinePrecision] * re + N[(-1.0 / im), $MachinePrecision]), $MachinePrecision] * re + N[(1.0 / im), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   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 51.5

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

   5. Applied rewrites 51.5%

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

   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 9.3

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

   8. Applied rewrites 9.3%

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

   10. Applied rewrites 9.3%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0

   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 98.9

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

   5. Applied rewrites 98.9%

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

   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 44.2%

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

   10. Applied rewrites 44.1%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 69.0%

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

   if -0.0 < (*.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 65.8

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

   5. Applied rewrites 65.8%

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

   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 55.7%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{im}, re, \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{\frac{1}{im} - \frac{re}{im}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 -0.1)
     (* (fma (* -0.16666666666666666 (* im im)) im im) (+ 1.0 re))
     (if (<= t_0 0.0)
       (/ 1.0 (- (/ 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 t_0 = sin(im) * exp(re);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = fma((-0.16666666666666666 * (im * im)), im, im) * (1.0 + re);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / ((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)
	t_0 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im) * Float64(1.0 + re));
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(1.0 / im) - Float64(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_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision] * N[(1.0 + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(N[(1.0 / im), $MachinePrecision] - N[(re / im), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   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 51.5

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

   5. Applied rewrites 51.5%

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

   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 9.3

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

   8. Applied rewrites 9.3%

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

   10. Applied rewrites 9.3%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0

   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 98.9

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

   5. Applied rewrites 98.9%

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

   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 44.2%

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

   10. Applied rewrites 44.1%

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

   11. Taylor expanded in re around 0

       \[\leadsto \frac{1}{-1 \cdot \frac{re}{im} + \frac{1}{\color{blue}{im}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 60.9%

       \[\leadsto \frac{1}{\frac{1}{im} - \frac{re}{\color{blue}{im}}} \]

   if -0.0 < (*.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 65.8

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

   5. Applied rewrites 65.8%

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

   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 55.7%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right) \cdot \left(1 + re\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 0:\\ \;\;\;\;\frac{1}{\frac{1}{im} - \frac{re}{im}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;e^{re} \leq 1.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re))))
   (if (<= (exp re) 0.0)
     t_0
     (if (<= (exp re) 1.002) (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)) t_0))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = t_0;
	} else if (exp(re) <= 1.002) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = t_0;
	elseif (exp(re) <= 1.002)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.002], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1.002 < (exp.f64 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 90.9

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

   5. Applied rewrites 90.9%

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

   if 0.0 < (exp.f64 re) < 1.002

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;e^{re} \leq 1.002:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re))))
   (if (<= (exp re) 5e-46)
     t_0
     (if (<= (exp re) 1.002) (* (+ 1.0 re) (sin im)) t_0))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double tmp;
	if (exp(re) <= 5e-46) {
		tmp = t_0;
	} else if (exp(re) <= 1.002) {
		tmp = (1.0 + re) * 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 (exp(re) <= 5d-46) then
        tmp = t_0
    else if (exp(re) <= 1.002d0) then
        tmp = (1.0d0 + re) * 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 (Math.exp(re) <= 5e-46) {
		tmp = t_0;
	} else if (Math.exp(re) <= 1.002) {
		tmp = (1.0 + re) * Math.sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * math.exp(re)
	tmp = 0
	if math.exp(re) <= 5e-46:
		tmp = t_0
	elif math.exp(re) <= 1.002:
		tmp = (1.0 + re) * math.sin(im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	tmp = 0.0
	if (exp(re) <= 5e-46)
		tmp = t_0;
	elseif (exp(re) <= 1.002)
		tmp = Float64(Float64(1.0 + re) * 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 (exp(re) <= 5e-46)
		tmp = t_0;
	elseif (exp(re) <= 1.002)
		tmp = (1.0 + re) * 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[N[Exp[re], $MachinePrecision], 5e-46], t$95$0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.002], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 4.99999999999999992e-46 or 1.002 < (exp.f64 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 90.2

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

   5. Applied rewrites 90.2%

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

   if 4.99999999999999992e-46 < (exp.f64 re) < 1.002

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \sin im \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.002:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;e^{re} \leq 1.002:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re))))
   (if (<= (exp re) 0.0) t_0 (if (<= (exp re) 1.002) (sin im) t_0))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = t_0;
	} else if (exp(re) <= 1.002) {
		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 (exp(re) <= 0.0d0) then
        tmp = t_0
    else if (exp(re) <= 1.002d0) 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 (Math.exp(re) <= 0.0) {
		tmp = t_0;
	} else if (Math.exp(re) <= 1.002) {
		tmp = Math.sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * math.exp(re)
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = t_0
	elif math.exp(re) <= 1.002:
		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 (exp(re) <= 0.0)
		tmp = t_0;
	elseif (exp(re) <= 1.002)
		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 (exp(re) <= 0.0)
		tmp = t_0;
	elseif (exp(re) <= 1.002)
		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[N[Exp[re], $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[Exp[re], $MachinePrecision], 1.002], N[Sin[im], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 1.002 < (exp.f64 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 90.9

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

   5. Applied rewrites 90.9%

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

   if 0.0 < (exp.f64 re) < 1.002

   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 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.002:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.0)
   (* (fma (* -0.16666666666666666 (* im im)) im im) (+ 1.0 re))
   (* (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)) <= 0.0) {
		tmp = fma((-0.16666666666666666 * (im * im)), im, im) * (1.0 + re);
	} 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)) <= 0.0)
		tmp = Float64(fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im) * Float64(1.0 + re));
	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], 0.0], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision] * N[(1.0 + re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 47.8

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

   5. Applied rewrites 47.8%

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

   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 28.4

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

   8. Applied rewrites 28.4%

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

   10. Applied rewrites 28.4%

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

   if -0.0 < (*.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 65.8

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

   5. Applied rewrites 65.8%

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

   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 55.7%

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

4. Final simplification 38.7%

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

Alternative 12: 28.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.975:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.975) (* 1.0 im) (* im re)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.975) {
		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 ((sin(im) * exp(re)) <= 0.975d0) 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 ((Math.sin(im) * Math.exp(re)) <= 0.975) {
		tmp = 1.0 * im;
	} else {
		tmp = im * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.sin(im) * math.exp(re)) <= 0.975:
		tmp = 1.0 * im
	else:
		tmp = im * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.975)
		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 ((sin(im) * exp(re)) <= 0.975)
		tmp = 1.0 * im;
	else
		tmp = im * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.975], N[(1.0 * im), $MachinePrecision], N[(im * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.974999999999999978

   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 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.0%

      \[\leadsto 1 \cdot im \]

   if 0.974999999999999978 < (*.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 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.5%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 23.4%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.975:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 93.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ \mathbf{if}\;re \leq -105:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 0.00192:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re))))
   (if (<= re -105.0)
     t_0
     (if (<= re 0.00192) (* (+ (fma (* re re) 0.5 1.0) re) (sin im)) t_0))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double tmp;
	if (re <= -105.0) {
		tmp = t_0;
	} else if (re <= 0.00192) {
		tmp = (fma((re * re), 0.5, 1.0) + re) * sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	tmp = 0.0
	if (re <= -105.0)
		tmp = t_0;
	elseif (re <= 0.00192)
		tmp = Float64(Float64(fma(Float64(re * re), 0.5, 1.0) + re) * sin(im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -105.0], t$95$0, If[LessEqual[re, 0.00192], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -105 or 0.00192000000000000005 < 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 90.9

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

   5. Applied rewrites 90.9%

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

   if -105 < re < 0.00192000000000000005

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -105:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 0.00192:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.5, 1\right) + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -16000000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{re}{im}, \mathsf{fma}\left(-0.16666666666666666, re, 0.5\right), \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{-8}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -16000000.0)
   (/
    1.0
    (fma
     (fma (/ re im) (fma -0.16666666666666666 re 0.5) (/ -1.0 im))
     re
     (/ 1.0 im)))
   (if (<= re 1.2e-8)
     (sin im)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -16000000.0) {
		tmp = 1.0 / fma(fma((re / im), fma(-0.16666666666666666, re, 0.5), (-1.0 / im)), re, (1.0 / im));
	} else if (re <= 1.2e-8) {
		tmp = sin(im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -16000000.0)
		tmp = Float64(1.0 / fma(fma(Float64(re / im), fma(-0.16666666666666666, re, 0.5), Float64(-1.0 / im)), re, Float64(1.0 / im)));
	elseif (re <= 1.2e-8)
		tmp = sin(im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -16000000.0], N[(1.0 / N[(N[(N[(re / im), $MachinePrecision] * N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] + N[(-1.0 / im), $MachinePrecision]), $MachinePrecision] * re + N[(1.0 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.2e-8], N[Sin[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 3 regimes

2. if re < -1.6e7

   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 100.0

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

   5. Applied rewrites 100.0%

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

   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 2.0%

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

   10. Applied rewrites 2.0%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 61.6%

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

   if -1.6e7 < re < 1.19999999999999999e-8

   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 97.4

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

   5. Applied rewrites 97.4%

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

   if 1.19999999999999999e-8 < 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 85.0

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

   5. Applied rewrites 85.0%

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

   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 59.7%

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

Alternative 15: 40.0% 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 70.8

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

5. Applied rewrites 70.8%

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

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 45.2%

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

Alternative 16: 38.4% 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 70.8

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

5. Applied rewrites 70.8%

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

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 43.4%

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

9. Final simplification 43.4%

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

Alternative 17: 37.9% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(N[(N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $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 70.8

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

5. Applied rewrites 70.8%

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

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 43.4%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 43.0%

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

12. Final simplification 43.0%

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

Alternative 18: 37.3% 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 70.8

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

5. Applied rewrites 70.8%

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

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 42.6%

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

Alternative 19: 34.4% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(N[(N[(im * re), $MachinePrecision] * 0.5 + 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 70.8

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

5. Applied rewrites 70.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 40.0%

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

Alternative 20: 29.8% accurate, 29.4× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (fma im re im))

C

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

Julia

function code(re, im)
	return fma(im, re, im)
end

Wolfram

code[re_, im_] := N[(im * 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 70.8

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

5. Applied rewrites 70.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 35.4%

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

Alternative 21: 6.9% 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 70.8

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

5. Applied rewrites 70.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 35.4%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 10.0%

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

Reproduce

herbie shell --seed 2024284 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))