math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 15.3s

Alternatives: 21

Speedup: 1.0×

• 24.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 93.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-231}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1 (* (sin im) (exp re)))
        (t_2 (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))))
   (if (<= t_1 (- INFINITY))
     (* (* (fma -0.16666666666666666 (* im im) 1.0) (exp re)) im)
     (if (<= t_1 -0.02)
       t_2
       (if (<= t_1 2e-231) t_0 (if (<= t_1 1.0) t_2 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = sin(im) * exp(re);
	double t_2 = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (im * im), 1.0) * exp(re)) * im;
	} else if (t_1 <= -0.02) {
		tmp = t_2;
	} else if (t_1 <= 2e-231) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = Float64(sin(im) * exp(re))
	t_2 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(im * im), 1.0) * exp(re)) * im);
	elseif (t_1 <= -0.02)
		tmp = t_2;
	elseif (t_1 <= 2e-231)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$2, If[LessEqual[t$95$1, 2e-231], t$95$0, If[LessEqual[t$95$1, 1.0], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 70.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 2e-231 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-231 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 2 \cdot 10^{-231}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, t\_3\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-231}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1 (* (sin im) (exp re)))
        (t_2 (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)))
        (t_3
         (fma
          (fma
           (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0)))
   (if (<= t_1 (- INFINITY))
     (* (fma (* t_3 (fma (fma 0.16666666666666666 re 0.5) re 1.0)) re t_3) im)
     (if (<= t_1 -0.02)
       t_2
       (if (<= t_1 2e-231) t_0 (if (<= t_1 1.0) t_2 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = sin(im) * exp(re);
	double t_2 = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	double t_3 = fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((t_3 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_3) * im;
	} else if (t_1 <= -0.02) {
		tmp = t_2;
	} else if (t_1 <= 2e-231) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = Float64(sin(im) * exp(re))
	t_2 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im))
	t_3 = fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(t_3 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_3) * im);
	elseif (t_1 <= -0.02)
		tmp = t_2;
	elseif (t_1 <= 2e-231)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t$95$3 * N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$3), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$2, If[LessEqual[t$95$1, 2e-231], t$95$0, If[LessEqual[t$95$1, 1.0], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 48.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 2e-231 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-231 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 2 \cdot 10^{-231}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ t_2 := \left(1 + re\right) \cdot \sin im\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, t\_3\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1 (* (sin im) (exp re)))
        (t_2 (* (+ 1.0 re) (sin im)))
        (t_3
         (fma
          (fma
           (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0)))
   (if (<= t_1 (- INFINITY))
     (* (fma (* t_3 (fma (fma 0.16666666666666666 re 0.5) re 1.0)) re t_3) im)
     (if (<= t_1 -0.02)
       t_2
       (if (<= t_1 5e-15) t_0 (if (<= t_1 1.0) t_2 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = sin(im) * exp(re);
	double t_2 = (1.0 + re) * sin(im);
	double t_3 = fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((t_3 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_3) * im;
	} else if (t_1 <= -0.02) {
		tmp = t_2;
	} else if (t_1 <= 5e-15) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = Float64(sin(im) * exp(re))
	t_2 = Float64(Float64(1.0 + re) * sin(im))
	t_3 = fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(t_3 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_3) * im);
	elseif (t_1 <= -0.02)
		tmp = t_2;
	elseif (t_1 <= 5e-15)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t$95$3 * N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$3), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$2, If[LessEqual[t$95$1, 5e-15], t$95$0, If[LessEqual[t$95$1, 1.0], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 48.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 4.99999999999999999e-15 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999999e-15 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 95.2

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

   5. Applied rewrites 95.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.02:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, t\_2\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1 (* (sin im) (exp re)))
        (t_2
         (fma
          (fma
           (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0)))
   (if (<= t_1 (- INFINITY))
     (* (fma (* t_2 (fma (fma 0.16666666666666666 re 0.5) re 1.0)) re t_2) im)
     (if (<= t_1 -0.02)
       (sin im)
       (if (<= t_1 5e-15) t_0 (if (<= t_1 1.0) (sin im) t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = sin(im) * exp(re);
	double t_2 = fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((t_2 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_2) * im;
	} else if (t_1 <= -0.02) {
		tmp = sin(im);
	} else if (t_1 <= 5e-15) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = Float64(sin(im) * exp(re))
	t_2 = fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(t_2 * fma(fma(0.16666666666666666, re, 0.5), re, 1.0)), re, t_2) * im);
	elseif (t_1 <= -0.02)
		tmp = sin(im);
	elseif (t_1 <= 5e-15)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t$95$2 * N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$2), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$1, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$1, 5e-15], t$95$0, If[LessEqual[t$95$1, 1.0], N[Sin[im], $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 48.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 4.99999999999999999e-15 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 96.0

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

   5. Applied rewrites 96.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999999e-15 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 95.2

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

   5. Applied rewrites 95.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(t$95$2 * t$95$1), $MachinePrecision] * re + t$95$2), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(N[(t$95$1 * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 48.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 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 66.7

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

   5. Applied rewrites 66.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 48.5%

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

4. Final simplification 61.9%

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

Alternative 7: 38.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(0.5, re, 1\right), re, t\_0\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0)))
   (if (<= (* (sin im) (exp re)) 5e-237)
     (* (fma (* t_0 (fma 0.5 re 1.0)) re t_0) im)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im))))

C

double code(double re, double im) {
	double t_0 = fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-237) {
		tmp = fma((t_0 * fma(0.5, re, 1.0)), re, t_0) * im;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-237)
		tmp = Float64(fma(Float64(t_0 * fma(0.5, re, 1.0)), re, t_0) * im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-237], N[(N[(N[(t$95$0 * N[(0.5 * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000002e-237

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.6%

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

   if 5.0000000000000002e-237 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.3%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(0.5, re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\left(\left(1 + re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-237)
   (*
    (*
     (+ 1.0 re)
     (fma
      (fma
       (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
       (* im im)
       -0.16666666666666666)
      (* im im)
      1.0))
    im)
   (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-237) {
		tmp = ((1.0 + re) * fma(fma(fma(-0.0001984126984126984, (im * im), 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0)) * im;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-237)
		tmp = Float64(Float64(Float64(1.0 + re) * fma(fma(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)) * im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-237], N[(N[(N[(1.0 + re), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000002e-237

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.5%

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

   if 5.0000000000000002e-237 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.3%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\left(\left(1 + re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.0001984126984126984, im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-237)
   (*
    (fma
     (fma (* (* im im) -0.0001984126984126984) (* im im) -0.16666666666666666)
     (* im im)
     1.0)
    im)
   (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-237) {
		tmp = fma(fma(((im * im) * -0.0001984126984126984), (im * im), -0.16666666666666666), (im * im), 1.0) * im;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-237)
		tmp = Float64(fma(fma(Float64(Float64(im * im) * -0.0001984126984126984), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0) * im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-237], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000002e-237

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 27.4%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040} \cdot {im}^{2}, im \cdot im, \frac{-1}{6}\right), im \cdot im, 1\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 27.4%

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

   if 5.0000000000000002e-237 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.3%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.0001984126984126984, im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-237)
   (fma (* (* im im) -0.16666666666666666) im im)
   (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-237) {
		tmp = fma(((im * im) * -0.16666666666666666), im, im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-237)
		tmp = fma(Float64(Float64(im * im) * -0.16666666666666666), im, im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-237], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000002e-237

   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 26.9%

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

   10. Applied rewrites 26.9%

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

   if 5.0000000000000002e-237 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.3%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot im, re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-237)
   (fma (* (* im im) -0.16666666666666666) im im)
   (fma (* (fma (fma 0.16666666666666666 re 0.5) re 1.0) im) re im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-237) {
		tmp = fma(((im * im) * -0.16666666666666666), im, im);
	} else {
		tmp = fma((fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-237)
		tmp = fma(Float64(Float64(im * im) * -0.16666666666666666), im, im);
	else
		tmp = fma(Float64(fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * im), re, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-237], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000002e-237

   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 26.9%

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

   10. Applied rewrites 26.9%

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

   if 5.0000000000000002e-237 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.3%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot im, re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot 0.16666666666666666\right) \cdot im, re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-237)
   (fma (* (* im im) -0.16666666666666666) im im)
   (fma (* (* (* re re) 0.16666666666666666) im) re im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-237) {
		tmp = fma(((im * im) * -0.16666666666666666), im, im);
	} else {
		tmp = fma((((re * re) * 0.16666666666666666) * im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-237)
		tmp = fma(Float64(Float64(im * im) * -0.16666666666666666), im, im);
	else
		tmp = fma(Float64(Float64(Float64(re * re) * 0.16666666666666666) * im), re, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-237], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000002e-237

   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 26.9%

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

   10. Applied rewrites 26.9%

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

   if 5.0000000000000002e-237 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 39.8%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot 0.16666666666666666\right) \cdot im, re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-15)
   (fma (* (* im im) -0.16666666666666666) im im)
   (* (* (* (fma 0.16666666666666666 re 0.5) re) im) re)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-15) {
		tmp = fma(((im * im) * -0.16666666666666666), im, im);
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-15)
		tmp = fma(Float64(Float64(im * im) * -0.16666666666666666), im, im);
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-15], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999999e-15

   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 46.2

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 35.0%

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

   10. Applied rewrites 35.0%

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

   if 4.99999999999999999e-15 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 27.1

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

   5. Applied rewrites 27.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 19.4%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 19.8%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-237)
   (fma (* (* im im) -0.16666666666666666) im im)
   (* (fma (fma 0.5 re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-237) {
		tmp = fma(((im * im) * -0.16666666666666666), im, im);
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-237)
		tmp = fma(Float64(Float64(im * im) * -0.16666666666666666), im, im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-237], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000002e-237

   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 26.9%

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

   10. Applied rewrites 26.9%

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

   if 5.0000000000000002e-237 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.3%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 32.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 5e-15)
   (fma (* (* im im) -0.16666666666666666) im im)
   (* (* (* re re) 0.5) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 5e-15) {
		tmp = fma(((im * im) * -0.16666666666666666), im, im);
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 5e-15)
		tmp = fma(Float64(Float64(im * im) * -0.16666666666666666), im, im);
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 5e-15], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im + im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999999e-15

   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 46.2

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 35.0%

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

   10. Applied rewrites 35.0%

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

   if 4.99999999999999999e-15 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 27.1

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

   5. Applied rewrites 27.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 19.4%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 19.9%

       \[\leadsto \left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 32.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.6) (fma re im im) (* (* (* re re) 0.5) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.6) {
		tmp = fma(re, im, im);
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.6)
		tmp = fma(re, im, im);
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.6], N[(re * im + im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.599999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 77.9

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.3%

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

   if 0.599999999999999978 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 34.6

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

   5. Applied rewrites 34.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 24.5%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.0%

       \[\leadsto \left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.47:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 800000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.47)
   (* im (exp re))
   (if (<= re 800000.0)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (sin im))
     (if (<= re 1.1e+103)
       (* (* (fma -0.16666666666666666 (* im im) 1.0) (exp re)) im)
       (* (* (* (fma 0.16666666666666666 re 0.5) re) re) (sin im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.47) {
		tmp = im * exp(re);
	} else if (re <= 800000.0) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	} else if (re <= 1.1e+103) {
		tmp = (fma(-0.16666666666666666, (im * im), 1.0) * exp(re)) * im;
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.47)
		tmp = Float64(im * exp(re));
	elseif (re <= 800000.0)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im));
	elseif (re <= 1.1e+103)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(im * im), 1.0) * exp(re)) * im);
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.47], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 800000.0], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.1e+103], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.46999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -0.46999999999999997 < re < 8e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 8e5 < re < 1.09999999999999996e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 87.5%

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

   if 1.09999999999999996e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.47:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 800000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.068:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 800000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.068)
   (* im (exp re))
   (if (<= re 800000.0)
     (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
     (if (<= re 1.1e+103)
       (* (* (fma -0.16666666666666666 (* im im) 1.0) (exp re)) im)
       (* (* (* (fma 0.16666666666666666 re 0.5) re) re) (sin im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.068) {
		tmp = im * exp(re);
	} else if (re <= 800000.0) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else if (re <= 1.1e+103) {
		tmp = (fma(-0.16666666666666666, (im * im), 1.0) * exp(re)) * im;
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.068)
		tmp = Float64(im * exp(re));
	elseif (re <= 800000.0)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	elseif (re <= 1.1e+103)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(im * im), 1.0) * exp(re)) * im);
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.068], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 800000.0], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.1e+103], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.068000000000000005

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -0.068000000000000005 < re < 8e5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 8e5 < re < 1.09999999999999996e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 87.5%

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

   if 1.09999999999999996e103 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.068:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 800000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 27.9% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.25 \cdot 10^{+16}:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 3.25e+16) (* 1.0 im) (* im re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.25e+16) {
		tmp = 1.0 * im;
	} else {
		tmp = im * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.25d+16) then
        tmp = 1.0d0 * im
    else
        tmp = im * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.25e+16) {
		tmp = 1.0 * im;
	} else {
		tmp = im * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.25e+16:
		tmp = 1.0 * im
	else:
		tmp = im * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.25e+16)
		tmp = Float64(1.0 * im);
	else
		tmp = Float64(im * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.25e+16)
		tmp = 1.0 * im;
	else
		tmp = im * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.25e+16], N[(1.0 * im), $MachinePrecision], N[(im * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.25e16

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 77.9

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.4%

      \[\leadsto 1 \cdot im \]

   if 3.25e16 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 7.0%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 7.4%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.25 \cdot 10^{+16}:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.3% accurate, 29.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 69.8

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

5. Applied rewrites 69.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 30.3%

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

Alternative 21: 6.8% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 69.8

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

5. Applied rewrites 69.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 30.3%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 7.6%

    \[\leadsto re \cdot im \]

12. Final simplification 7.6%

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

Reproduce

herbie shell --seed 2024268 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))