math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 11.3s

Alternatives: 20

Speedup: 1.0×

• 25.4% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 93.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right) \cdot \left(im \cdot e^{re}\right)\\ \mathbf{elif}\;t\_0 \leq -0.01 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (fma (* -0.16666666666666666 im) im 1.0) (* im (exp re)))
     (if (or (<= t_0 -0.01) (not (or (<= t_0 0.0) (not (<= t_0 1.0)))))
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((-0.16666666666666666 * im), im, 1.0) * (im * exp(re));
	} else if ((t_0 <= -0.01) || !((t_0 <= 0.0) || !(t_0 <= 1.0))) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(-0.16666666666666666 * im), im, 1.0) * Float64(im * exp(re)));
	elseif ((t_0 <= -0.01) || !((t_0 <= 0.0) || !(t_0 <= 1.0)))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.01], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + {im}^{2} \cdot \left(\frac{-1}{6} \cdot e^{re} + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot e^{re}\right) + \frac{1}{120} \cdot e^{re}\right)\right)\right)} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot e^{re}\right) \cdot im} \]

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 82.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 92.7

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

   5. Applied rewrites 92.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right) \cdot \left(im \cdot e^{re}\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01 \lor \neg \left(e^{re} \cdot \sin im \leq 0 \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right)\\ t_1 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_0, im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(t\_0 \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -0.01 \lor \neg \left(t\_1 \leq 0 \lor \neg \left(t\_1 \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im)
          -0.16666666666666666))
        (t_1 (* (exp re) (sin im))))
   (if (<= t_1 (- INFINITY))
     (*
      (+
       (fma
        (fma
         (* (fma t_0 (* im im) 1.0) re)
         (fma 0.16666666666666666 re 0.5)
         1.0)
        re
        (* (- re -1.0) (* (* t_0 im) im)))
       1.0)
      im)
     (if (or (<= t_1 -0.01) (not (or (<= t_1 0.0) (not (<= t_1 1.0)))))
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666);
	double t_1 = exp(re) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma((fma(t_0, (im * im), 1.0) * re), fma(0.16666666666666666, re, 0.5), 1.0), re, ((re - -1.0) * ((t_0 * im) * im))) + 1.0) * im;
	} else if ((t_1 <= -0.01) || !((t_1 <= 0.0) || !(t_1 <= 1.0))) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666)
	t_1 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(fma(t_0, Float64(im * im), 1.0) * re), fma(0.16666666666666666, re, 0.5), 1.0), re, Float64(Float64(re - -1.0) * Float64(Float64(t_0 * im) * im))) + 1.0) * im);
	elseif ((t_1 <= -0.01) || !((t_1 <= 0.0) || !(t_1 <= 1.0)))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(N[(t$95$0 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * re + N[(N[(re - -1.0), $MachinePrecision] * N[(N[(t$95$0 * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision], If[Or[LessEqual[t$95$1, -0.01], N[Not[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + {im}^{2} \cdot \left(\frac{-1}{6} \cdot e^{re} + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot e^{re}\right) + \frac{1}{120} \cdot e^{re}\right)\right)\right)} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot e^{re}\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(1 + \left(re \cdot \left(1 + \left(re \cdot \left(\frac{1}{6} \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + \frac{1}{2} \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) \cdot im \]

   8. Applied rewrites 66.1%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 92.7

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

   5. Applied rewrites 92.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01 \lor \neg \left(e^{re} \cdot \sin im \leq 0 \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right) \cdot \left(im \cdot e^{re}\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (fma (* -0.16666666666666666 im) im 1.0) (* im (exp re)))
     (if (<= t_0 -0.01)
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (sin im))
       (if (or (<= t_0 0.0) (not (<= t_0 1.0)))
         (* (exp re) im)
         (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((-0.16666666666666666 * im), im, 1.0) * (im * exp(re));
	} else if (t_0 <= -0.01) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	} else if ((t_0 <= 0.0) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(-0.16666666666666666 * im), im, 1.0) * Float64(im * exp(re)));
	elseif (t_0 <= -0.01)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im));
	elseif ((t_0 <= 0.0) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + {im}^{2} \cdot \left(\frac{-1}{6} \cdot e^{re} + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot e^{re}\right) + \frac{1}{120} \cdot e^{re}\right)\right)\right)} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot e^{re}\right) \cdot im} \]

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 82.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002

   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 99.1

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

   5. Applied rewrites 99.1%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right) \cdot \left(im \cdot e^{re}\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0 \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right)\\ t_1 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_0, im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(t\_0 \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -0.01 \lor \neg \left(t\_1 \leq 2 \cdot 10^{-142} \lor \neg \left(t\_1 \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im)
          -0.16666666666666666))
        (t_1 (* (exp re) (sin im))))
   (if (<= t_1 (- INFINITY))
     (*
      (+
       (fma
        (fma
         (* (fma t_0 (* im im) 1.0) re)
         (fma 0.16666666666666666 re 0.5)
         1.0)
        re
        (* (- re -1.0) (* (* t_0 im) im)))
       1.0)
      im)
     (if (or (<= t_1 -0.01) (not (or (<= t_1 2e-142) (not (<= t_1 1.0)))))
       (* (+ 1.0 re) (sin im))
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666);
	double t_1 = exp(re) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma((fma(t_0, (im * im), 1.0) * re), fma(0.16666666666666666, re, 0.5), 1.0), re, ((re - -1.0) * ((t_0 * im) * im))) + 1.0) * im;
	} else if ((t_1 <= -0.01) || !((t_1 <= 2e-142) || !(t_1 <= 1.0))) {
		tmp = (1.0 + re) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666)
	t_1 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(fma(t_0, Float64(im * im), 1.0) * re), fma(0.16666666666666666, re, 0.5), 1.0), re, Float64(Float64(re - -1.0) * Float64(Float64(t_0 * im) * im))) + 1.0) * im);
	elseif ((t_1 <= -0.01) || !((t_1 <= 2e-142) || !(t_1 <= 1.0)))
		tmp = Float64(Float64(1.0 + re) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(N[(t$95$0 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * re + N[(N[(re - -1.0), $MachinePrecision] * N[(N[(t$95$0 * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision], If[Or[LessEqual[t$95$1, -0.01], N[Not[Or[LessEqual[t$95$1, 2e-142], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + {im}^{2} \cdot \left(\frac{-1}{6} \cdot e^{re} + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot e^{re}\right) + \frac{1}{120} \cdot e^{re}\right)\right)\right)} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot e^{re}\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(1 + \left(re \cdot \left(1 + \left(re \cdot \left(\frac{1}{6} \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + \frac{1}{2} \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) \cdot im \]

   8. Applied rewrites 66.1%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 2.0000000000000001e-142 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000001e-142 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.5

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

   5. Applied rewrites 93.5%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01 \lor \neg \left(e^{re} \cdot \sin im \leq 2 \cdot 10^{-142} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right)\\ t_1 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_0, im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(t\_0 \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -0.01 \lor \neg \left(t\_1 \leq 5 \cdot 10^{-98} \lor \neg \left(t\_1 \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im)
          -0.16666666666666666))
        (t_1 (* (exp re) (sin im))))
   (if (<= t_1 (- INFINITY))
     (*
      (+
       (fma
        (fma
         (* (fma t_0 (* im im) 1.0) re)
         (fma 0.16666666666666666 re 0.5)
         1.0)
        re
        (* (- re -1.0) (* (* t_0 im) im)))
       1.0)
      im)
     (if (or (<= t_1 -0.01) (not (or (<= t_1 5e-98) (not (<= t_1 1.0)))))
       (sin im)
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666);
	double t_1 = exp(re) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma((fma(t_0, (im * im), 1.0) * re), fma(0.16666666666666666, re, 0.5), 1.0), re, ((re - -1.0) * ((t_0 * im) * im))) + 1.0) * im;
	} else if ((t_1 <= -0.01) || !((t_1 <= 5e-98) || !(t_1 <= 1.0))) {
		tmp = sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666)
	t_1 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(fma(t_0, Float64(im * im), 1.0) * re), fma(0.16666666666666666, re, 0.5), 1.0), re, Float64(Float64(re - -1.0) * Float64(Float64(t_0 * im) * im))) + 1.0) * im);
	elseif ((t_1 <= -0.01) || !((t_1 <= 5e-98) || !(t_1 <= 1.0)))
		tmp = sin(im);
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(N[(t$95$0 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * re + N[(N[(re - -1.0), $MachinePrecision] * N[(N[(t$95$0 * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision], If[Or[LessEqual[t$95$1, -0.01], N[Not[Or[LessEqual[t$95$1, 5e-98], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[Sin[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + {im}^{2} \cdot \left(\frac{-1}{6} \cdot e^{re} + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot e^{re}\right) + \frac{1}{120} \cdot e^{re}\right)\right)\right)} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot e^{re}\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(1 + \left(re \cdot \left(1 + \left(re \cdot \left(\frac{1}{6} \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + \frac{1}{2} \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) \cdot im \]

   8. Applied rewrites 66.1%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 5.00000000000000018e-98 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5.00000000000000018e-98 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.9

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

   5. Applied rewrites 93.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01 \lor \neg \left(e^{re} \cdot \sin im \leq 5 \cdot 10^{-98} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ t_1 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot t\_0, re, t\_0\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-98}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, re, 1\right), \frac{re}{im}, \frac{-1}{im}\right), re, {im}^{-1}\right)\right)}^{-1}\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0))
        (t_1 (* (exp re) (sin im))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* (fma 0.5 re 1.0) t_0) re t_0) im)
     (if (<= t_1 -0.01)
       (sin im)
       (if (<= t_1 5e-98)
         (pow
          (fma (fma (fma -1.0 re 1.0) (/ re im) (/ -1.0 im)) re (pow im -1.0))
          -1.0)
         (if (<= t_1 1.0)
           (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 t_0 = fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double t_1 = exp(re) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((fma(0.5, re, 1.0) * t_0), re, t_0) * im;
	} else if (t_1 <= -0.01) {
		tmp = sin(im);
	} else if (t_1 <= 5e-98) {
		tmp = pow(fma(fma(fma(-1.0, re, 1.0), (re / im), (-1.0 / im)), re, pow(im, -1.0)), -1.0);
	} else if (t_1 <= 1.0) {
		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)
	t_0 = fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	t_1 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(fma(0.5, re, 1.0) * t_0), re, t_0) * im);
	elseif (t_1 <= -0.01)
		tmp = sin(im);
	elseif (t_1 <= 5e-98)
		tmp = fma(fma(fma(-1.0, re, 1.0), Float64(re / im), Float64(-1.0 / im)), re, (im ^ -1.0)) ^ -1.0;
	elseif (t_1 <= 1.0)
		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_] := Block[{t$95$0 = N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$1, -0.01], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$1, 5e-98], N[Power[N[(N[(N[(-1.0 * re + 1.0), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[(-1.0 / im), $MachinePrecision]), $MachinePrecision] * re + N[Power[im, -1.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$1, 1.0], 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 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 im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + {im}^{2} \cdot \left(\frac{-1}{6} \cdot e^{re} + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot e^{re}\right) + \frac{1}{120} \cdot e^{re}\right)\right)\right)} \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot e^{re}\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(1 + \left(re \cdot \left(1 + \left(\frac{1}{2} \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 59.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 5.00000000000000018e-98 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5.00000000000000018e-98

   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

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

   8. Applied rewrites 45.7%

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

   10. Applied rewrites 45.6%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 84.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 11.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 11.6%

       \[\leadsto im \cdot re \]

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 48.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-98}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, re, 1\right), \frac{re}{im}, \frac{-1}{im}\right), re, {im}^{-1}\right)\right)}^{-1}\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, re, 1\right), \frac{re}{im}, \frac{-1}{im}\right), re, {im}^{-1}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.01)
     (*
      (fma (fma 0.5 re 1.0) re 1.0)
      (fma (* im im) (* im -0.16666666666666666) im))
     (if (<= t_0 0.0)
       (pow
        (fma (fma (fma -1.0 re 1.0) (/ re im) (/ -1.0 im)) re (pow im -1.0))
        -1.0)
       (* (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 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma((im * im), (im * -0.16666666666666666), im);
	} else if (t_0 <= 0.0) {
		tmp = pow(fma(fma(fma(-1.0, re, 1.0), (re / im), (-1.0 / im)), re, pow(im, -1.0)), -1.0);
	} 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(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), Float64(im * -0.16666666666666666), im));
	elseif (t_0 <= 0.0)
		tmp = fma(fma(fma(-1.0, re, 1.0), Float64(re / im), Float64(-1.0 / im)), re, (im ^ -1.0)) ^ -1.0;
	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[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Power[N[(N[(N[(-1.0 * re + 1.0), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[(-1.0 / im), $MachinePrecision]), $MachinePrecision] * re + N[Power[im, -1.0], $MachinePrecision]), $MachinePrecision], -1.0], $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)) < -0.0100000000000000002

   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 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 29.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 29.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.9%

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

   if -0.0100000000000000002 < (*.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 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

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

   8. Applied rewrites 32.2%

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

   10. Applied rewrites 32.1%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 80.3%

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

   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 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 6.5%

       \[\leadsto im \cdot re \]

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 43.9%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, re, 1\right), \frac{re}{im}, \frac{-1}{im}\right), re, {im}^{-1}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{re}{im}, re + -1, {im}^{-1}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.01)
     (*
      (fma (fma 0.5 re 1.0) re 1.0)
      (fma (* im im) (* im -0.16666666666666666) im))
     (if (<= t_0 0.0)
       (pow (fma (/ re im) (+ re -1.0) (pow im -1.0)) -1.0)
       (* (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 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma((im * im), (im * -0.16666666666666666), im);
	} else if (t_0 <= 0.0) {
		tmp = pow(fma((re / im), (re + -1.0), pow(im, -1.0)), -1.0);
	} 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(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), Float64(im * -0.16666666666666666), im));
	elseif (t_0 <= 0.0)
		tmp = fma(Float64(re / im), Float64(re + -1.0), (im ^ -1.0)) ^ -1.0;
	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[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Power[N[(N[(re / im), $MachinePrecision] * N[(re + -1.0), $MachinePrecision] + N[Power[im, -1.0], $MachinePrecision]), $MachinePrecision], -1.0], $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)) < -0.0100000000000000002

   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 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 29.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 29.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.9%

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

   if -0.0100000000000000002 < (*.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 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

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

   8. Applied rewrites 32.2%

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

   10. Applied rewrites 32.1%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 74.0%

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

   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 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 6.5%

       \[\leadsto im \cdot re \]

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 43.9%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{re}{im}, re + -1, {im}^{-1}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{\left({im}^{-1} - \frac{re}{im}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.01)
     (*
      (fma (fma 0.5 re 1.0) re 1.0)
      (fma (* im im) (* im -0.16666666666666666) im))
     (if (<= t_0 0.0)
       (pow (- (pow im -1.0) (/ re im)) -1.0)
       (* (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 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma((im * im), (im * -0.16666666666666666), im);
	} else if (t_0 <= 0.0) {
		tmp = pow((pow(im, -1.0) - (re / im)), -1.0);
	} 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(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), Float64(im * -0.16666666666666666), im));
	elseif (t_0 <= 0.0)
		tmp = Float64((im ^ -1.0) - Float64(re / im)) ^ -1.0;
	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[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Power[N[(N[Power[im, -1.0], $MachinePrecision] - N[(re / im), $MachinePrecision]), $MachinePrecision], -1.0], $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)) < -0.0100000000000000002

   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 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 29.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 29.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.9%

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

   if -0.0100000000000000002 < (*.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 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

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

   8. Applied rewrites 32.2%

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

   10. Applied rewrites 32.1%

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

   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 58.2%

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

   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 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 6.5%

       \[\leadsto im \cdot re \]

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 43.9%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;{\left({im}^{-1} - \frac{re}{im}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right)\\ \mathbf{if}\;re \leq -5.2 \cdot 10^{-16}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, re, 1\right), \frac{re}{im}, \frac{-1}{im}\right), re, {im}^{-1}\right)\right)}^{-1}\\ \mathbf{elif}\;re \leq 3100000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_0, im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(t\_0 \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im)
          -0.16666666666666666)))
   (if (<= re -5.2e-16)
     (pow
      (fma (fma (fma -1.0 re 1.0) (/ re im) (/ -1.0 im)) re (pow im -1.0))
      -1.0)
     (if (<= re 3100000.0)
       (sin im)
       (*
        (+
         (fma
          (fma
           (* (fma t_0 (* im im) 1.0) re)
           (fma 0.16666666666666666 re 0.5)
           1.0)
          re
          (* (- re -1.0) (* (* t_0 im) im)))
         1.0)
        im)))))

C

double code(double re, double im) {
	double t_0 = fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666);
	double tmp;
	if (re <= -5.2e-16) {
		tmp = pow(fma(fma(fma(-1.0, re, 1.0), (re / im), (-1.0 / im)), re, pow(im, -1.0)), -1.0);
	} else if (re <= 3100000.0) {
		tmp = sin(im);
	} else {
		tmp = (fma(fma((fma(t_0, (im * im), 1.0) * re), fma(0.16666666666666666, re, 0.5), 1.0), re, ((re - -1.0) * ((t_0 * im) * im))) + 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666)
	tmp = 0.0
	if (re <= -5.2e-16)
		tmp = fma(fma(fma(-1.0, re, 1.0), Float64(re / im), Float64(-1.0 / im)), re, (im ^ -1.0)) ^ -1.0;
	elseif (re <= 3100000.0)
		tmp = sin(im);
	else
		tmp = Float64(Float64(fma(fma(Float64(fma(t_0, Float64(im * im), 1.0) * re), fma(0.16666666666666666, re, 0.5), 1.0), re, Float64(Float64(re - -1.0) * Float64(Float64(t_0 * im) * im))) + 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]}, If[LessEqual[re, -5.2e-16], N[Power[N[(N[(N[(-1.0 * re + 1.0), $MachinePrecision] * N[(re / im), $MachinePrecision] + N[(-1.0 / im), $MachinePrecision]), $MachinePrecision] * re + N[Power[im, -1.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[re, 3100000.0], N[Sin[im], $MachinePrecision], N[(N[(N[(N[(N[(N[(t$95$0 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * re + N[(N[(re - -1.0), $MachinePrecision] * N[(N[(t$95$0 * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.1999999999999997e-16

   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

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

   8. Applied rewrites 5.5%

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

   10. Applied rewrites 5.5%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 72.8%

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

   if -5.1999999999999997e-16 < re < 3.1e6

   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 96.7

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

   5. Applied rewrites 96.7%

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

   if 3.1e6 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(e^{re} + {im}^{2} \cdot \left(\frac{-1}{6} \cdot e^{re} + {im}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot e^{re}\right) + \frac{1}{120} \cdot e^{re}\right)\right)\right)} \]

   5. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot e^{re}\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(1 + \left(re \cdot \left(1 + \left(re \cdot \left(\frac{1}{6} \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + \frac{1}{2} \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right) \cdot im \]

   8. Applied rewrites 62.5%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{-16}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, re, 1\right), \frac{re}{im}, \frac{-1}{im}\right), re, {im}^{-1}\right)\right)}^{-1}\\ \mathbf{elif}\;re \leq 3100000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right), re, \left(re - -1\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im\right) \cdot im\right)\right) + 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 5e-8)
   (*
    (fma (fma 0.5 re 1.0) re 1.0)
    (fma (* im im) (* im -0.16666666666666666) im))
   (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 5e-8) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma((im * im), (im * -0.16666666666666666), im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 5e-8)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), Float64(im * -0.16666666666666666), im));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision] + im), $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)) < 4.9999999999999998e-8

   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 57.7

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 42.1

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 42.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.1%

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

   if 4.9999999999999998e-8 < (*.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 32.4

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 6.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 7.1%

       \[\leadsto im \cdot re \]

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 22.2%

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

Alternative 13: 34.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{im}{\mathsf{fma}\left(-0.6666666666666666, re, 2\right)}, re, im\right), re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.75)
   (fma (fma (/ im (fma -0.6666666666666666 re 2.0)) re im) re im)
   (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.75) {
		tmp = fma(fma((im / fma(-0.6666666666666666, re, 2.0)), re, im), re, im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.75)
		tmp = fma(fma(Float64(im / fma(-0.6666666666666666, re, 2.0)), re, im), re, im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.75], N[(N[(N[(im / N[(-0.6666666666666666 * re + 2.0), $MachinePrecision]), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.75

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.1%

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

   10. Applied rewrites 35.1%

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

   11. Taylor expanded in re around 0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{im}{2 + \frac{-2}{3} \cdot re}, re, im\right), re, im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 31.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{im}{\mathsf{fma}\left(-0.6666666666666666, re, 2\right)}, re, im\right), re, im\right) \]

   if 0.75 < (*.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 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 8.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 8.5%

       \[\leadsto im \cdot re \]

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 30.2%

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

Alternative 14: 33.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.95:\\ \;\;\;\;\mathsf{fma}\left(im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.95)
   (fma im re im)
   (* (* (* (fma 0.16666666666666666 re 0.5) re) im) re)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.95) {
		tmp = fma(im, re, im);
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.95)
		tmp = fma(im, re, im);
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.95], N[(im * re + im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.94999999999999996

   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 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.1%

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

   if 0.94999999999999996 < (*.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 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.3%

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

   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 30.4%

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

   12. Taylor expanded in re around inf

       \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im + \color{blue}{\frac{1}{2} \cdot \frac{im}{re}}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 30.7%

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

Alternative 15: 39.4% accurate, 8.6× speedup

Math

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

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 65.7

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

5. Applied rewrites 65.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 25.9%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 5.1%

    \[\leadsto im \cdot re \]

12. Taylor expanded in re around 0

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

14. Applied rewrites 35.2%

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

Alternative 16: 37.6% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * im), $MachinePrecision]), $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 65.7

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

5. Applied rewrites 65.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 32.6%

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

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 33.0%

    \[\leadsto \mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot 0.16666666666666666\right) \cdot im, re, im\right) \]
12. Step-by-step derivation

13. Applied rewrites 33.0%

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

Alternative 17: 37.1% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(N[(N[(im * re), $MachinePrecision] * N[(0.16666666666666666 * re), $MachinePrecision]), $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 65.7

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

5. Applied rewrites 65.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 32.6%

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

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 33.0%

    \[\leadsto \mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot 0.16666666666666666\right) \cdot im, re, im\right) \]
12. Step-by-step derivation

13. Applied rewrites 32.3%

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

Alternative 18: 34.0% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 65.7

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

5. Applied rewrites 65.7%

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

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 29.0%

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

Alternative 19: 29.7% 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 65.7

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

5. Applied rewrites 65.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 25.9%

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

Alternative 20: 7.1% 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 65.7

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

5. Applied rewrites 65.7%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 25.9%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 5.1%

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

Reproduce

herbie shell --seed 2024308 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))