math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 16.5s

Alternatives: 22

Speedup: 1.0×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ t_2 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ t_3 := \sin im \cdot t\_2\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (* (exp re) im))
        (t_2 (fma re (fma re 0.5 1.0) 1.0))
        (t_3 (* (sin im) t_2)))
   (if (<= t_0 (- INFINITY))
     (* t_2 (fma im (* im (* im -0.16666666666666666)) im))
     (if (<= t_0 -0.01)
       t_3
       (if (<= t_0 5e-106) t_1 (if (<= t_0 1.0) t_3 t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double t_2 = fma(re, fma(re, 0.5, 1.0), 1.0);
	double t_3 = sin(im) * t_2;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_2 * fma(im, (im * (im * -0.16666666666666666)), im);
	} else if (t_0 <= -0.01) {
		tmp = t_3;
	} else if (t_0 <= 5e-106) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	t_2 = fma(re, fma(re, 0.5, 1.0), 1.0)
	t_3 = Float64(sin(im) * t_2)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_2 * fma(im, Float64(im * Float64(im * -0.16666666666666666)), im));
	elseif (t_0 <= -0.01)
		tmp = t_3;
	elseif (t_0 <= 5e-106)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$2 = N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[im], $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$2 * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], t$95$3, If[LessEqual[t$95$0, 5e-106], t$95$1, If[LessEqual[t$95$0, 1.0], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 4.99999999999999983e-106 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999983e-106 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 96.1

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

   5. Applied rewrites 96.1%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-106}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;t\_0 \leq 1.4 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma im (* im (* im -0.16666666666666666)) im))
     (if (<= t_0 -0.01)
       (* (sin im) (+ re 1.0))
       (if (<= t_0 1.4e-61) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, (im * (im * -0.16666666666666666)), im);
	} else if (t_0 <= -0.01) {
		tmp = sin(im) * (re + 1.0);
	} else if (t_0 <= 1.4e-61) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, Float64(im * Float64(im * -0.16666666666666666)), im));
	elseif (t_0 <= -0.01)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	elseif (t_0 <= 1.4e-61)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.4e-61], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.4000000000000001e-61 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if 1.4000000000000001e-61 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1.4 \cdot 10^{-61}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 1.4 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma im (* im (* im -0.16666666666666666)) im))
     (if (<= t_0 -0.01)
       (sin im)
       (if (<= t_0 1.4e-61) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, (im * (im * -0.16666666666666666)), im);
	} else if (t_0 <= -0.01) {
		tmp = sin(im);
	} else if (t_0 <= 1.4e-61) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, Float64(im * Float64(im * -0.16666666666666666)), im));
	elseif (t_0 <= -0.01)
		tmp = sin(im);
	elseif (t_0 <= 1.4e-61)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1.4e-61], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 1.4000000000000001e-61 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.4000000000000001e-61 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 95.7

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

   5. Applied rewrites 95.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1.4 \cdot 10^{-61}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 1.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{im}, \mathsf{fma}\left(re, 0.5, -1\right), \frac{re \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)}{im}\right), \frac{1}{im}\right)}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma im (* im (* im -0.16666666666666666)) im))
     (if (<= t_0 -0.01)
       (sin im)
       (if (<= t_0 1.4e-61)
         (/
          1.0
          (fma
           re
           (fma (/ 1.0 im) (fma re 0.5 -1.0) (/ (* re (* (* re re) -0.25)) im))
           (/ 1.0 im)))
         (if (<= t_0 1.0)
           (sin im)
           (* im (* 0.16666666666666666 (* re (* re re))))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, (im * (im * -0.16666666666666666)), im);
	} else if (t_0 <= -0.01) {
		tmp = sin(im);
	} else if (t_0 <= 1.4e-61) {
		tmp = 1.0 / fma(re, fma((1.0 / im), fma(re, 0.5, -1.0), ((re * ((re * re) * -0.25)) / im)), (1.0 / im));
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, Float64(im * Float64(im * -0.16666666666666666)), im));
	elseif (t_0 <= -0.01)
		tmp = sin(im);
	elseif (t_0 <= 1.4e-61)
		tmp = Float64(1.0 / fma(re, fma(Float64(1.0 / im), fma(re, 0.5, -1.0), Float64(Float64(re * Float64(Float64(re * re) * -0.25)) / im)), Float64(1.0 / im)));
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1.4e-61], N[(1.0 / N[(re * N[(N[(1.0 / im), $MachinePrecision] * N[(re * 0.5 + -1.0), $MachinePrecision] + N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision] + N[(1.0 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(im * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 1.4000000000000001e-61 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.4000000000000001e-61

   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. lower-*.f64 N/A

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

      2. lower-exp.f64 99.3

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

   5. Applied rewrites 99.3%

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

   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 50.2%

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

   10. Applied rewrites 50.1%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 87.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 56.1%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 56.1%

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

Alternative 6: 58.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot \left(im \cdot -0.16666666666666666\right), im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(re, \mathsf{fma}\left(\frac{1}{im}, \mathsf{fma}\left(re, 0.5, -1\right), \frac{re \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)}{im}\right), \frac{1}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.01)
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma im (* im (* im -0.16666666666666666)) im))
     (if (<= t_0 0.0)
       (/
        1.0
        (fma
         re
         (fma (/ 1.0 im) (fma re 0.5 -1.0) (/ (* re (* (* re re) -0.25)) im))
         (/ 1.0 im)))
       (* im (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, (im * (im * -0.16666666666666666)), im);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / fma(re, fma((1.0 / im), fma(re, 0.5, -1.0), ((re * ((re * re) * -0.25)) / im)), (1.0 / im));
	} else {
		tmp = im * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im, Float64(im * Float64(im * -0.16666666666666666)), im));
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 / fma(re, fma(Float64(1.0 / im), fma(re, 0.5, -1.0), Float64(Float64(re * Float64(Float64(re * re) * -0.25)) / im)), Float64(1.0 / im)));
	else
		tmp = Float64(im * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(re * N[(N[(1.0 / im), $MachinePrecision] * N[(re * 0.5 + -1.0), $MachinePrecision] + N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision] + N[(1.0 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 80.3

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

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 25.3

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

   8. Applied rewrites 25.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 34.1%

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

   10. Applied rewrites 34.1%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 83.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.4%

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

Alternative 7: 51.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.01)
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma im (* im (* im -0.16666666666666666)) im))
     (if (<= t_0 0.0)
       (/ 1.0 (fma re (* (/ 1.0 im) (fma re 0.5 -1.0)) (/ 1.0 im)))
       (* im (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))))

C

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

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(re * N[(N[(1.0 / im), $MachinePrecision] * N[(re * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 80.3

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

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 25.3

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

   8. Applied rewrites 25.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 34.1%

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

   10. Applied rewrites 34.1%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 65.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.4%

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

Alternative 8: 45.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.01)
     (*
      (fma re (fma re 0.5 1.0) 1.0)
      (fma im (* im (* im -0.16666666666666666)) im))
     (if (<= t_0 0.0)
       (/ 1.0 (- (/ 1.0 im) (/ re im)))
       (* im (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))))

C

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

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(N[(1.0 / im), $MachinePrecision] - N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 80.3

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

   5. Applied rewrites 80.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 25.3

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

   8. Applied rewrites 25.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 34.1%

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

   10. Applied rewrites 34.1%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 53.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.4%

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

Alternative 9: 36.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 27.8

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

   8. Applied rewrites 27.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.4%

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

4. Final simplification 39.8%

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

Alternative 10: 35.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $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 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 28.6%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 27.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.4%

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

Alternative 11: 35.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0005)
   (fma im (* -0.16666666666666666 (* im im)) im)
   (* im (* re (* re (fma re 0.16666666666666666 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0005) {
		tmp = fma(im, (-0.16666666666666666 * (im * im)), im);
	} else {
		tmp = im * (re * (re * fma(re, 0.16666666666666666, 0.5)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0005)
		tmp = fma(im, Float64(-0.16666666666666666 * Float64(im * im)), im);
	else
		tmp = Float64(im * Float64(re * Float64(re * fma(re, 0.16666666666666666, 0.5))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0005], N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000001e-4

   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 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 41.4%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 40.1%

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

   if 5.0000000000000001e-4 < (*.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. lower-*.f64 N/A

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

      2. lower-exp.f64 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 34.5%

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

Alternative 12: 35.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0005)
   (fma im (* -0.16666666666666666 (* im im)) im)
   (* im (* 0.16666666666666666 (* re (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0005) {
		tmp = fma(im, (-0.16666666666666666 * (im * im)), im);
	} else {
		tmp = im * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0005)
		tmp = fma(im, Float64(-0.16666666666666666 * Float64(im * im)), im);
	else
		tmp = Float64(im * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0005], N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000001e-4

   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 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 41.4%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 40.1%

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

   if 5.0000000000000001e-4 < (*.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. lower-*.f64 N/A

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

      2. lower-exp.f64 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 34.7%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 34.7%

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

Alternative 13: 34.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(im * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $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 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 28.6%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 27.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 59.1%

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

Alternative 14: 34.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0005)
   (fma im (* -0.16666666666666666 (* im im)) im)
   (* 0.5 (* im (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0005) {
		tmp = fma(im, (-0.16666666666666666 * (im * im)), im);
	} else {
		tmp = 0.5 * (im * (re * re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0005)
		tmp = fma(im, Float64(-0.16666666666666666 * Float64(im * im)), im);
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0005], N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision], N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000001e-4

   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 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 41.4%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 40.1%

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

   if 5.0000000000000001e-4 < (*.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. lower-*.f64 N/A

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

      2. lower-exp.f64 48.4

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

   5. Applied rewrites 48.4%

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

   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 24.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 28.5%

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

Alternative 15: 32.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.98:\\ \;\;\;\;im \cdot 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.98) (* im 1.0) (* 0.5 (* im (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.98) {
		tmp = im * 1.0;
	} else {
		tmp = 0.5 * (im * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * sin(im)) <= 0.98d0) then
        tmp = im * 1.0d0
    else
        tmp = 0.5d0 * (im * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= 0.98) {
		tmp = im * 1.0;
	} else {
		tmp = 0.5 * (im * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 0.98:
		tmp = im * 1.0
	else:
		tmp = 0.5 * (im * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.98)
		tmp = Float64(im * 1.0);
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 0.98)
		tmp = im * 1.0;
	else
		tmp = 0.5 * (im * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.98], N[(im * 1.0), $MachinePrecision], N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.97999999999999998

   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. lower-*.f64 N/A

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

      2. lower-exp.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.0%

      \[\leadsto im \cdot 1 \]

   if 0.97999999999999998 < (*.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. lower-*.f64 N/A

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

      2. lower-exp.f64 79.3

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

   5. Applied rewrites 79.3%

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

   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 39.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 45.9%

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

Alternative 16: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), -re\right)\\ \mathbf{if}\;re \leq -0.07:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right) \cdot t\_0, \frac{1}{t\_0}, 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma re (* re (fma re 0.16666666666666666 0.5)) (- re))))
   (if (<= re -0.07)
     (* (exp re) im)
     (if (<= re 350.0)
       (*
        (sin im)
        (fma
         (* (fma (fma re 0.16666666666666666 0.5) (* re re) re) t_0)
         (/ 1.0 t_0)
         1.0))
       (if (<= re 1e+103)
         (* (exp re) (fma im (* -0.16666666666666666 (* im im)) im))
         (* (sin im) (* re (* re (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double t_0 = fma(re, (re * fma(re, 0.16666666666666666, 0.5)), -re);
	double tmp;
	if (re <= -0.07) {
		tmp = exp(re) * im;
	} else if (re <= 350.0) {
		tmp = sin(im) * fma((fma(fma(re, 0.16666666666666666, 0.5), (re * re), re) * t_0), (1.0 / t_0), 1.0);
	} else if (re <= 1e+103) {
		tmp = exp(re) * fma(im, (-0.16666666666666666 * (im * im)), im);
	} else {
		tmp = sin(im) * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(re, Float64(re * fma(re, 0.16666666666666666, 0.5)), Float64(-re))
	tmp = 0.0
	if (re <= -0.07)
		tmp = Float64(exp(re) * im);
	elseif (re <= 350.0)
		tmp = Float64(sin(im) * fma(Float64(fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), re) * t_0), Float64(1.0 / t_0), 1.0));
	elseif (re <= 1e+103)
		tmp = Float64(exp(re) * fma(im, Float64(-0.16666666666666666 * Float64(im * im)), im));
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + (-re)), $MachinePrecision]}, If[LessEqual[re, -0.07], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 350.0], N[(N[Sin[im], $MachinePrecision] * N[(N[(N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + re), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.070000000000000007

   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. lower-*.f64 N/A

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

      2. lower-exp.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -0.070000000000000007 < re < 350

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.2%

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

   if 350 < re < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if 1e103 < 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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.07:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right) \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), -re\right), \frac{1}{\mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), -re\right)}, 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.07:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.07)
   (* (exp re) im)
   (if (<= re 350.0)
     (* (sin im) (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))
     (if (<= re 1e+103)
       (* (exp re) (fma im (* -0.16666666666666666 (* im im)) im))
       (* (sin im) (* re (* re (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.07) {
		tmp = exp(re) * im;
	} else if (re <= 350.0) {
		tmp = sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (re <= 1e+103) {
		tmp = exp(re) * fma(im, (-0.16666666666666666 * (im * im)), im);
	} else {
		tmp = sin(im) * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.07)
		tmp = Float64(exp(re) * im);
	elseif (re <= 350.0)
		tmp = Float64(sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	elseif (re <= 1e+103)
		tmp = Float64(exp(re) * fma(im, Float64(-0.16666666666666666 * Float64(im * im)), im));
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.07], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 350.0], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.070000000000000007

   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. lower-*.f64 N/A

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

      2. lower-exp.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -0.070000000000000007 < re < 350

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 350 < re < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if 1e103 < 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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.07:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-5}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.15e-5)
   (* (exp re) im)
   (if (<= re 350.0)
     (* (sin im) (fma re (fma re 0.5 1.0) 1.0))
     (if (<= re 1e+103)
       (* (exp re) (fma im (* -0.16666666666666666 (* im im)) im))
       (* (sin im) (* re (* re (* re 0.16666666666666666))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.15e-5) {
		tmp = exp(re) * im;
	} else if (re <= 350.0) {
		tmp = sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else if (re <= 1e+103) {
		tmp = exp(re) * fma(im, (-0.16666666666666666 * (im * im)), im);
	} else {
		tmp = sin(im) * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.15e-5)
		tmp = Float64(exp(re) * im);
	elseif (re <= 350.0)
		tmp = Float64(sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0));
	elseif (re <= 1e+103)
		tmp = Float64(exp(re) * fma(im, Float64(-0.16666666666666666 * Float64(im * im)), im));
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.15e-5], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 350.0], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.1500000000000001e-5

   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. lower-*.f64 N/A

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

      2. lower-exp.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -2.1500000000000001e-5 < re < 350

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 350 < re < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if 1e103 < 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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-5}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 350:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 96.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -2.15 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 5000000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -2.15e-5)
     t_0
     (if (<= re 5000000.0)
       (* (sin im) (fma re (fma re 0.5 1.0) 1.0))
       (if (<= re 1e+103)
         t_0
         (* (sin im) (* re (* re (* re 0.16666666666666666)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -2.15e-5) {
		tmp = t_0;
	} else if (re <= 5000000.0) {
		tmp = sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else if (re <= 1e+103) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (re * (re * (re * 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -2.15e-5)
		tmp = t_0;
	elseif (re <= 5000000.0)
		tmp = Float64(sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0));
	elseif (re <= 1e+103)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(re * Float64(re * Float64(re * 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -2.15e-5], t$95$0, If[LessEqual[re, 5000000.0], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+103], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.1500000000000001e-5 or 5e6 < re < 1e103

   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. lower-*.f64 N/A

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

      2. lower-exp.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if -2.1500000000000001e-5 < re < 5e6

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 1e103 < 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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{-5}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 5000000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;re \leq 10^{+103}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 28.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2 \cdot 10^{+71}:\\ \;\;\;\;im \cdot 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 6.2e+71) (* im 1.0) (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.2e+71) {
		tmp = im * 1.0;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.2d+71) then
        tmp = im * 1.0d0
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.2e+71) {
		tmp = im * 1.0;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.2e+71:
		tmp = im * 1.0
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.2e+71)
		tmp = Float64(im * 1.0);
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.2e+71)
		tmp = im * 1.0;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.2e+71], N[(im * 1.0), $MachinePrecision], N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.20000000000000036e71

   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. lower-*.f64 N/A

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

      2. lower-exp.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 37.7%

      \[\leadsto im \cdot 1 \]

   if 6.20000000000000036e71 < 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. lower-*.f64 N/A

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

      2. lower-exp.f64 35.1

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 11.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 12.9%

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

Alternative 21: 30.1% 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. lower-*.f64 N/A

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

   2. lower-exp.f64 70.1

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

5. Applied rewrites 70.1%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 33.3%

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

Alternative 22: 6.6% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return re * im;
}

Python

def code(re, im):
	return re * im

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(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. lower-*.f64 N/A

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

   2. lower-exp.f64 70.1

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

5. Applied rewrites 70.1%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 33.3%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 7.7%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))