math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 20

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re \cdot re, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (* (* (fma -0.16666666666666666 (* im im) 1.0) (exp re)) im)
     (if (<= t_0 -5e-24)
       (* (fma (fma 0.16666666666666666 re 0.5) (* re re) (+ 1.0 re)) (sin im))
       (if (<= t_0 1e-83)
         t_1
         (if (<= t_0 1000.0)
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (sin im))
           t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (im * im), 1.0) * exp(re)) * im;
	} else if (t_0 <= -5e-24) {
		tmp = fma(fma(0.16666666666666666, re, 0.5), (re * re), (1.0 + re)) * sin(im);
	} else if (t_0 <= 1e-83) {
		tmp = t_1;
	} else if (t_0 <= 1000.0) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(im * im), 1.0) * exp(re)) * im);
	elseif (t_0 <= -5e-24)
		tmp = Float64(fma(fma(0.16666666666666666, re, 0.5), Float64(re * re), Float64(1.0 + re)) * sin(im));
	elseif (t_0 <= 1e-83)
		tmp = t_1;
	elseif (t_0 <= 1000.0)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, -5e-24], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(1.0 + re), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-83], t$95$1, If[LessEqual[t$95$0, 1000.0], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 82.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 82.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -4.9999999999999998e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   if -4.9999999999999998e-24 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-83 or 1e3 < (*.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 94.9

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

   5. Applied rewrites 94.9%

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

   if 1e-83 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 93.9

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

   5. Applied rewrites 93.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re \cdot re, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-83}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1 (* (sin im) (exp re)))
        (t_2
         (*
          (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
          (sin im))))
   (if (<= t_1 (- INFINITY))
     (* (* (fma -0.16666666666666666 (* im im) 1.0) (exp re)) im)
     (if (<= t_1 -5e-24)
       t_2
       (if (<= t_1 1e-83) t_0 (if (<= t_1 1000.0) t_2 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = sin(im) * exp(re);
	double t_2 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (im * im), 1.0) * exp(re)) * im;
	} else if (t_1 <= -5e-24) {
		tmp = t_2;
	} else if (t_1 <= 1e-83) {
		tmp = t_0;
	} else if (t_1 <= 1000.0) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = Float64(sin(im) * exp(re))
	t_2 = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(im * im), 1.0) * exp(re)) * im);
	elseif (t_1 <= -5e-24)
		tmp = t_2;
	elseif (t_1 <= 1e-83)
		tmp = t_0;
	elseif (t_1 <= 1000.0)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$1, -5e-24], t$95$2, If[LessEqual[t$95$1, 1e-83], t$95$0, If[LessEqual[t$95$1, 1000.0], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 82.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 82.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -4.9999999999999998e-24 or 1e-83 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if -4.9999999999999998e-24 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-83 or 1e3 < (*.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 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-83}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (* (* (fma -0.16666666666666666 (* im im) 1.0) (exp re)) im)
     (if (<= t_0 -5e-24)
       (* (fma (* re re) 0.5 (+ 1.0 re)) (sin im))
       (if (<= t_0 1e-83)
         t_1
         (if (<= t_0 1000.0)
           (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
           t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (im * im), 1.0) * exp(re)) * im;
	} else if (t_0 <= -5e-24) {
		tmp = fma((re * re), 0.5, (1.0 + re)) * sin(im);
	} else if (t_0 <= 1e-83) {
		tmp = t_1;
	} else if (t_0 <= 1000.0) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(im * im), 1.0) * exp(re)) * im);
	elseif (t_0 <= -5e-24)
		tmp = Float64(fma(Float64(re * re), 0.5, Float64(1.0 + re)) * sin(im));
	elseif (t_0 <= 1e-83)
		tmp = t_1;
	elseif (t_0 <= 1000.0)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, -5e-24], N[(N[(N[(re * re), $MachinePrecision] * 0.5 + N[(1.0 + re), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-83], t$95$1, If[LessEqual[t$95$0, 1000.0], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 82.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 82.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -4.9999999999999998e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   if -4.9999999999999998e-24 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-83 or 1e3 < (*.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 94.9

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

   5. Applied rewrites 94.9%

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

   if 1e-83 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 93.9

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

   5. Applied rewrites 93.9%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-83}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ t_1 := \sin im \cdot e^{re}\\ t_2 := im \cdot e^{re}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, t\_0\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;t\_1 \leq 10^{-83}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* (sin im) (exp re)))
        (t_2 (* im (exp re))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* t_0 (fma (fma 0.16666666666666666 re 0.5) re 1.0)) re t_0) im)
     (if (<= t_1 -5e-24)
       (* (fma (* re re) 0.5 (+ 1.0 re)) (sin im))
       (if (<= t_1 1e-83)
         t_2
         (if (<= t_1 1000.0)
           (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
           t_2))))))

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 = sin(im) * exp(re);
	double t_2 = im * exp(re);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((t_0 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_0) * im;
	} else if (t_1 <= -5e-24) {
		tmp = fma((re * re), 0.5, (1.0 + re)) * sin(im);
	} else if (t_1 <= 1e-83) {
		tmp = t_2;
	} else if (t_1 <= 1000.0) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = t_2;
	}
	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(sin(im) * exp(re))
	t_2 = Float64(im * exp(re))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(t_0 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_0) * im);
	elseif (t_1 <= -5e-24)
		tmp = Float64(fma(Float64(re * re), 0.5, Float64(1.0 + re)) * sin(im));
	elseif (t_1 <= 1e-83)
		tmp = t_2;
	elseif (t_1 <= 1000.0)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = t_2;
	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[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t$95$0 * N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$1, -5e-24], N[(N[(N[(re * re), $MachinePrecision] * 0.5 + N[(1.0 + re), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-83], t$95$2, If[LessEqual[t$95$1, 1000.0], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

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 82.4%

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

   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%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -4.9999999999999998e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   if -4.9999999999999998e-24 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-83 or 1e3 < (*.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 94.9

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

   5. Applied rewrites 94.9%

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

   if 1e-83 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 93.9

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

   5. Applied rewrites 93.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-83}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, t\_3\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1 (* (sin im) (exp re)))
        (t_2 (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)))
        (t_3
         (fma
          (fma
           (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0)))
   (if (<= t_1 (- INFINITY))
     (* (fma (* t_3 (fma (fma 0.16666666666666666 re 0.5) re 1.0)) re t_3) im)
     (if (<= t_1 -5e-24)
       t_2
       (if (<= t_1 1e-83) t_0 (if (<= t_1 1000.0) t_2 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = sin(im) * exp(re);
	double t_2 = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	double t_3 = fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((t_3 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_3) * im;
	} else if (t_1 <= -5e-24) {
		tmp = t_2;
	} else if (t_1 <= 1e-83) {
		tmp = t_0;
	} else if (t_1 <= 1000.0) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = Float64(sin(im) * exp(re))
	t_2 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im))
	t_3 = fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(t_3 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_3) * im);
	elseif (t_1 <= -5e-24)
		tmp = t_2;
	elseif (t_1 <= 1e-83)
		tmp = t_0;
	elseif (t_1 <= 1000.0)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t$95$3 * N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$3), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$1, -5e-24], t$95$2, If[LessEqual[t$95$1, 1e-83], t$95$0, If[LessEqual[t$95$1, 1000.0], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 82.4%

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

   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%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -4.9999999999999998e-24 or 1e-83 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if -4.9999999999999998e-24 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-83 or 1e3 < (*.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 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-83}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, t\_2\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{elif}\;t\_1 \leq 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1 (* (sin im) (exp re)))
        (t_2
         (fma
          (fma
           (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0)))
   (if (<= t_1 (- INFINITY))
     (* (fma (* t_2 (fma (fma 0.16666666666666666 re 0.5) re 1.0)) re t_2) im)
     (if (<= t_1 -5e-24)
       (* (+ 1.0 re) (sin im))
       (if (<= t_1 1e-83) t_0 (if (<= t_1 1000.0) (sin im) t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = sin(im) * exp(re);
	double t_2 = fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((t_2 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_2) * im;
	} else if (t_1 <= -5e-24) {
		tmp = (1.0 + re) * sin(im);
	} else if (t_1 <= 1e-83) {
		tmp = t_0;
	} else if (t_1 <= 1000.0) {
		tmp = sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = Float64(sin(im) * exp(re))
	t_2 = fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(t_2 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_2) * im);
	elseif (t_1 <= -5e-24)
		tmp = Float64(Float64(1.0 + re) * sin(im));
	elseif (t_1 <= 1e-83)
		tmp = t_0;
	elseif (t_1 <= 1000.0)
		tmp = sin(im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t$95$2 * N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$2), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$1, -5e-24], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-83], t$95$0, If[LessEqual[t$95$1, 1000.0], N[Sin[im], $MachinePrecision], t$95$0]]]]]]]

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 82.4%

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

   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%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -4.9999999999999998e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -4.9999999999999998e-24 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-83 or 1e3 < (*.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 94.9

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

   5. Applied rewrites 94.9%

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

   if 1e-83 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 91.7

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

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\sin im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-83}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, t\_2\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_1 \leq 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1 (* (sin im) (exp re)))
        (t_2
         (fma
          (fma
           (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0)))
   (if (<= t_1 (- INFINITY))
     (* (fma (* t_2 (fma (fma 0.16666666666666666 re 0.5) re 1.0)) re t_2) im)
     (if (<= t_1 -5e-24)
       (sin im)
       (if (<= t_1 1e-83) t_0 (if (<= t_1 1000.0) (sin im) t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = sin(im) * exp(re);
	double t_2 = fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((t_2 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_2) * im;
	} else if (t_1 <= -5e-24) {
		tmp = sin(im);
	} else if (t_1 <= 1e-83) {
		tmp = t_0;
	} else if (t_1 <= 1000.0) {
		tmp = sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = Float64(sin(im) * exp(re))
	t_2 = fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(t_2 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_2) * im);
	elseif (t_1 <= -5e-24)
		tmp = sin(im);
	elseif (t_1 <= 1e-83)
		tmp = t_0;
	elseif (t_1 <= 1000.0)
		tmp = sin(im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t$95$2 * N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$2), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$1, -5e-24], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$1, 1e-83], t$95$0, If[LessEqual[t$95$1, 1000.0], N[Sin[im], $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 82.4%

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

   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%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -4.9999999999999998e-24 or 1e-83 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if -4.9999999999999998e-24 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-83 or 1e3 < (*.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 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -5 \cdot 10^{-24}:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-83}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, t\_1\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 1000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re)))
        (t_1
         (fma
          (fma
           (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0)))
   (if (<= t_0 (- INFINITY))
     (* (fma (* t_1 (fma (fma 0.16666666666666666 re 0.5) re 1.0)) re t_1) im)
     (if (<= t_0 1000.0)
       (sin im)
       (fma (fma (* (fma 0.16666666666666666 re 0.5) im) re im) re im)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((t_1 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_1) * im;
	} else if (t_0 <= 1000.0) {
		tmp = sin(im);
	} else {
		tmp = fma(fma((fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(t_1 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_1) * im);
	elseif (t_0 <= 1000.0)
		tmp = sin(im);
	else
		tmp = fma(fma(Float64(fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(t$95$1 * N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$1), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 1000.0], N[Sin[im], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision] * re + 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 82.4%

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

   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%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e3

   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 64.3

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

   5. Applied rewrites 64.3%

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

   if 1e3 < (*.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 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 52.7%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1000:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.05:\\ \;\;\;\;\left(\left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.05)
   (*
    (*
     (- re -1.0)
     (fma
      (fma
       (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
       (* im im)
       -0.16666666666666666)
      (* im im)
      1.0))
    im)
   (fma (fma (* (fma 0.16666666666666666 re 0.5) im) re im) re im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.05) {
		tmp = ((re - -1.0) * fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0)) * im;
	} else {
		tmp = fma(fma((fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.05)
		tmp = Float64(Float64(Float64(re - -1.0) * fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)) * im);
	else
		tmp = fma(fma(Float64(fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(N[(re - -1.0), $MachinePrecision] * 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]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.050000000000000003

   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 61.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 32.7%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 35.8

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

   5. Applied rewrites 35.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.4%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.05:\\ \;\;\;\;\left(\left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.0)
   (*
    (fma
     (fma
      (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
      (* im im)
      -0.16666666666666666)
     (* im im)
     1.0)
    im)
   (fma (fma (* (fma 0.16666666666666666 re 0.5) im) re im) re im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.0) {
		tmp = fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0) * im;
	} else {
		tmp = fma(fma((fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.0)
		tmp = Float64(fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0) * im);
	else
		tmp = fma(fma(Float64(fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. 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 56.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.5%

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

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

   1. Initial program 99.9%

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

   3. 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 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.9%

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

4. Final simplification 30.2%

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

Alternative 12: 34.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 41.8

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

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 22.5%

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

   10. Applied rewrites 22.5%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.9%

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

4. Final simplification 29.6%

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

Alternative 13: 33.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 41.8

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

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 22.5%

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

   10. Applied rewrites 22.5%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 51.3

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

   5. Applied rewrites 51.3%

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

   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 38.5%

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

4. Final simplification 28.0%

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

Alternative 14: 30.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 41.8

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

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 22.5%

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

   10. Applied rewrites 22.5%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 31.0%

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

4. Final simplification 25.4%

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

Alternative 15: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + re}\\ \mathbf{if}\;re \leq -54:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 0.036:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re \cdot re, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+68}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), t\_0, 0.16666666666666666 \cdot re\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot t\_0} \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 re))))
   (if (<= re -54.0)
     (* im (exp re))
     (if (<= re 0.036)
       (* (fma (fma 0.16666666666666666 re 0.5) (* re re) (+ 1.0 re)) (sin im))
       (if (<= re 3e+68)
         (* (* (fma -0.16666666666666666 (* im im) 1.0) (exp re)) im)
         (*
          (/
           (fma
            (* (fma 0.027777777777777776 (* re re) -0.25) (* re re))
            t_0
            (* 0.16666666666666666 re))
           (* (fma re 0.16666666666666666 -0.5) t_0))
          (sin im)))))))

C

double code(double re, double im) {
	double t_0 = 1.0 / (1.0 + re);
	double tmp;
	if (re <= -54.0) {
		tmp = im * exp(re);
	} else if (re <= 0.036) {
		tmp = fma(fma(0.16666666666666666, re, 0.5), (re * re), (1.0 + re)) * sin(im);
	} else if (re <= 3e+68) {
		tmp = (fma(-0.16666666666666666, (im * im), 1.0) * exp(re)) * im;
	} else {
		tmp = (fma((fma(0.027777777777777776, (re * re), -0.25) * (re * re)), t_0, (0.16666666666666666 * re)) / (fma(re, 0.16666666666666666, -0.5) * t_0)) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(1.0 / Float64(1.0 + re))
	tmp = 0.0
	if (re <= -54.0)
		tmp = Float64(im * exp(re));
	elseif (re <= 0.036)
		tmp = Float64(fma(fma(0.16666666666666666, re, 0.5), Float64(re * re), Float64(1.0 + re)) * sin(im));
	elseif (re <= 3e+68)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(im * im), 1.0) * exp(re)) * im);
	else
		tmp = Float64(Float64(fma(Float64(fma(0.027777777777777776, Float64(re * re), -0.25) * Float64(re * re)), t_0, Float64(0.16666666666666666 * re)) / Float64(fma(re, 0.16666666666666666, -0.5) * t_0)) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -54.0], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.036], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(1.0 + re), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3e+68], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(0.027777777777777776 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision] / N[(N[(re * 0.16666666666666666 + -0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -54

   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%

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

   if -54 < re < 0.0359999999999999973

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.6%

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

   if 0.0359999999999999973 < re < 3.0000000000000002e68

   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 82.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 82.4%

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

   if 3.0000000000000002e68 < re

   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 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 86.6%

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

   9. Applied rewrites 98.1%

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

   10. Taylor expanded in re around inf

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{36}, re \cdot re, \frac{-1}{4}\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \frac{1}{6} \cdot re\right)}{\mathsf{fma}\left(re, \frac{1}{6}, \color{blue}{\frac{-1}{2}}\right) \cdot \frac{1}{1 + re}} \cdot \sin im \]
   11. Step-by-step derivation

   12. Applied rewrites 98.1%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, 0.16666666666666666 \cdot re\right)}{\mathsf{fma}\left(re, 0.16666666666666666, \color{blue}{-0.5}\right) \cdot \frac{1}{1 + re}} \cdot \sin im \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -54:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 0.036:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re \cdot re, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+68}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, 0.16666666666666666 \cdot re\right)}{\mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot \frac{1}{1 + re}} \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;\sin im \leq 5 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, t\_0\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \end{array} \]

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)))
   (if (<= (sin im) 5e-74)
     (* (fma (* t_0 (fma (fma 0.16666666666666666 re 0.5) re 1.0)) re t_0) im)
     (fma (fma (* (fma 0.16666666666666666 re 0.5) im) re im) re 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 tmp;
	if (sin(im) <= 5e-74) {
		tmp = fma((t_0 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_0) * im;
	} else {
		tmp = fma(fma((fma(0.16666666666666666, re, 0.5) * im), re, im), re, 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)
	tmp = 0.0
	if (sin(im) <= 5e-74)
		tmp = Float64(fma(Float64(t_0 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_0) * im);
	else
		tmp = fma(fma(Float64(fma(0.16666666666666666, re, 0.5) * im), re, im), re, 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]}, If[LessEqual[N[Sin[im], $MachinePrecision], 5e-74], N[(N[(N[(t$95$0 * N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 im) < 4.99999999999999998e-74

   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 66.7%

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

   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 43.9%

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

   if 4.99999999999999998e-74 < (sin.f64 im)

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 50.2

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

   5. Applied rewrites 50.2%

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

   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 18.7%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \leq 5 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;\sin im \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(0.5, re, 1\right), re, t\_0\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \end{array} \]

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)))
   (if (<= (sin im) 2e-289)
     (* (fma (* t_0 (fma 0.5 re 1.0)) re t_0) im)
     (fma (fma (* (fma 0.16666666666666666 re 0.5) im) re im) re 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 tmp;
	if (sin(im) <= 2e-289) {
		tmp = fma((t_0 * fma(0.5, re, 1.0)), re, t_0) * im;
	} else {
		tmp = fma(fma((fma(0.16666666666666666, re, 0.5) * im), re, im), re, 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)
	tmp = 0.0
	if (sin(im) <= 2e-289)
		tmp = Float64(fma(Float64(t_0 * fma(0.5, re, 1.0)), re, t_0) * im);
	else
		tmp = fma(fma(Float64(fma(0.16666666666666666, re, 0.5) * im), re, im), re, 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]}, If[LessEqual[N[Sin[im], $MachinePrecision], 2e-289], N[(N[(N[(t$95$0 * N[(0.5 * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 im) < 2e-289

   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 59.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 37.7%

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

   if 2e-289 < (sin.f64 im)

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 63.2

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

   5. Applied rewrites 63.2%

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

   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 30.2%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(0.5, re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -54:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 0.036:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re \cdot re, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+68}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)\right)}{\mathsf{fma}\left(0.6666666666666666, re, -0.5\right)} \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -54.0)
   (* im (exp re))
   (if (<= re 0.036)
     (* (fma (fma 0.16666666666666666 re 0.5) (* re re) (+ 1.0 re)) (sin im))
     (if (<= re 3e+68)
       (* (* (fma -0.16666666666666666 (* im im) 1.0) (exp re)) im)
       (*
        (/
         (fma
          (* (fma 0.027777777777777776 (* re re) -0.25) (* re re))
          (/ 1.0 (+ 1.0 re))
          (fma re 0.16666666666666666 -0.5))
         (fma 0.6666666666666666 re -0.5))
        (sin im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -54.0) {
		tmp = im * exp(re);
	} else if (re <= 0.036) {
		tmp = fma(fma(0.16666666666666666, re, 0.5), (re * re), (1.0 + re)) * sin(im);
	} else if (re <= 3e+68) {
		tmp = (fma(-0.16666666666666666, (im * im), 1.0) * exp(re)) * im;
	} else {
		tmp = (fma((fma(0.027777777777777776, (re * re), -0.25) * (re * re)), (1.0 / (1.0 + re)), fma(re, 0.16666666666666666, -0.5)) / fma(0.6666666666666666, re, -0.5)) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -54.0)
		tmp = Float64(im * exp(re));
	elseif (re <= 0.036)
		tmp = Float64(fma(fma(0.16666666666666666, re, 0.5), Float64(re * re), Float64(1.0 + re)) * sin(im));
	elseif (re <= 3e+68)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(im * im), 1.0) * exp(re)) * im);
	else
		tmp = Float64(Float64(fma(Float64(fma(0.027777777777777776, Float64(re * re), -0.25) * Float64(re * re)), Float64(1.0 / Float64(1.0 + re)), fma(re, 0.16666666666666666, -0.5)) / fma(0.6666666666666666, re, -0.5)) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -54.0], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 0.036], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + N[(1.0 + re), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3e+68], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(0.027777777777777776 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + re), $MachinePrecision]), $MachinePrecision] + N[(re * 0.16666666666666666 + -0.5), $MachinePrecision]), $MachinePrecision] / N[(0.6666666666666666 * re + -0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -54

   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%

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

   if -54 < re < 0.0359999999999999973

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.6%

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

   if 0.0359999999999999973 < re < 3.0000000000000002e68

   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 82.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 82.4%

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

   if 3.0000000000000002e68 < re

   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 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 86.6%

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

   9. Applied rewrites 98.1%

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

   10. Taylor expanded in re around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{36}, re \cdot re, \frac{-1}{4}\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, \frac{1}{6}, \frac{-1}{2}\right) \cdot 1\right)}{\frac{2}{3} \cdot re - \color{blue}{\frac{1}{2}}} \cdot \sin im \]
   11. Step-by-step derivation

   12. Applied rewrites 98.1%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right) \cdot 1\right)}{\mathsf{fma}\left(0.6666666666666666, \color{blue}{re}, -0.5\right)} \cdot \sin im \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -54:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 0.036:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re \cdot re, 1 + re\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 3 \cdot 10^{+68}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.027777777777777776, re \cdot re, -0.25\right) \cdot \left(re \cdot re\right), \frac{1}{1 + re}, \mathsf{fma}\left(re, 0.16666666666666666, -0.5\right)\right)}{\mathsf{fma}\left(0.6666666666666666, re, -0.5\right)} \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.4% 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 64.1

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

5. Applied rewrites 64.1%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 24.5%

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

Alternative 20: 6.6% 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 64.1

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

5. Applied rewrites 64.1%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 24.5%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 6.4%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))