math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 10.8s

Alternatives: 16

Speedup: 1.0×

• 24.7% 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: 93.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{elif}\;t\_1 \leq -0.005:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -0.005)
       t_2
       (if (<= t_1 1e-136) 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 <= -0.005) {
		tmp = t_2;
	} else if (t_1 <= 1e-136) {
		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 <= -0.005)
		tmp = t_2;
	elseif (t_1 <= 1e-136)
		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, -0.005], t$95$2, If[LessEqual[t$95$1, 1e-136], 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 73.7%

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

   6. Taylor expanded in 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 73.7%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0050000000000000001 or 1e-136 < (*.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 97.9

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

   5. Applied rewrites 97.9%

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

   if -0.0050000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-136 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-136}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ t_3 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot t\_2, re, t\_2\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq -0.005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-136}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re)))
        (t_1 (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)))
        (t_2
         (fma
          (fma
           (fma (* im im) -0.0001984126984126984 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0))
        (t_3 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (* (fma (* (fma (fma 0.16666666666666666 re 0.5) re 1.0) t_2) re t_2) im)
     (if (<= t_0 -0.005)
       t_1
       (if (<= t_0 1e-136) t_3 (if (<= t_0 1.0) t_1 t_3))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	double t_2 = fma(fma(fma((im * im), -0.0001984126984126984, 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double t_3 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * t_2), re, t_2) * im;
	} else if (t_0 <= -0.005) {
		tmp = t_1;
	} else if (t_0 <= 1e-136) {
		tmp = t_3;
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im))
	t_2 = fma(fma(fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	t_3 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * t_2), re, t_2) * im);
	elseif (t_0 <= -0.005)
		tmp = t_1;
	elseif (t_0 <= 1e-136)
		tmp = t_3;
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] * re + t$95$2), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, -0.005], t$95$1, If[LessEqual[t$95$0, 1e-136], t$95$3, If[LessEqual[t$95$0, 1.0], t$95$1, t$95$3]]]]]]]]

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 73.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0050000000000000001 or 1e-136 < (*.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 97.9

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

   5. Applied rewrites 97.9%

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

   if -0.0050000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-136 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-136}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ t_2 := \sin im \cdot e^{re}\\ t_3 := \left(1 + re\right) \cdot \sin im\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot t\_1, re, t\_1\right) \cdot im\\ \mathbf{elif}\;t\_2 \leq -0.005:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1
         (fma
          (fma
           (fma (* im im) -0.0001984126984126984 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0))
        (t_2 (* (sin im) (exp re)))
        (t_3 (* (+ 1.0 re) (sin im))))
   (if (<= t_2 (- INFINITY))
     (* (fma (* (fma (fma 0.16666666666666666 re 0.5) re 1.0) t_1) re t_1) im)
     (if (<= t_2 -0.005)
       t_3
       (if (<= t_2 1e-136) t_0 (if (<= t_2 1.0) t_3 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = fma(fma(fma((im * im), -0.0001984126984126984, 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double t_2 = sin(im) * exp(re);
	double t_3 = (1.0 + re) * sin(im);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * t_1), re, t_1) * im;
	} else if (t_2 <= -0.005) {
		tmp = t_3;
	} else if (t_2 <= 1e-136) {
		tmp = t_0;
	} else if (t_2 <= 1.0) {
		tmp = t_3;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = fma(fma(fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	t_2 = Float64(sin(im) * exp(re))
	t_3 = Float64(Float64(1.0 + re) * sin(im))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(Float64(fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * t_1), re, t_1) * im);
	elseif (t_2 <= -0.005)
		tmp = t_3;
	elseif (t_2 <= 1e-136)
		tmp = t_0;
	elseif (t_2 <= 1.0)
		tmp = t_3;
	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[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * re + t$95$1), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$2, -0.005], t$95$3, If[LessEqual[t$95$2, 1e-136], t$95$0, If[LessEqual[t$95$2, 1.0], t$95$3, 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 73.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0050000000000000001 or 1e-136 < (*.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 97.5

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

   5. Applied rewrites 97.5%

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

   if -0.0050000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-136 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.005:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-136}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ t_2 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot t\_1, re, t\_1\right) \cdot im\\ \mathbf{elif}\;t\_2 \leq -0.005:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_2 \leq 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1
         (fma
          (fma
           (fma (* im im) -0.0001984126984126984 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0))
        (t_2 (* (sin im) (exp re))))
   (if (<= t_2 (- INFINITY))
     (* (fma (* (fma (fma 0.16666666666666666 re 0.5) re 1.0) t_1) re t_1) im)
     (if (<= t_2 -0.005)
       (sin im)
       (if (<= t_2 1e-136) t_0 (if (<= t_2 1.0) (sin im) t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = fma(fma(fma((im * im), -0.0001984126984126984, 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double t_2 = sin(im) * exp(re);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * t_1), re, t_1) * im;
	} else if (t_2 <= -0.005) {
		tmp = sin(im);
	} else if (t_2 <= 1e-136) {
		tmp = t_0;
	} else if (t_2 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = fma(fma(fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	t_2 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(Float64(fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * t_1), re, t_1) * im);
	elseif (t_2 <= -0.005)
		tmp = sin(im);
	elseif (t_2 <= 1e-136)
		tmp = t_0;
	elseif (t_2 <= 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[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * re + t$95$1), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$2, -0.005], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$2, 1e-136], t$95$0, If[LessEqual[t$95$2, 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 73.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0050000000000000001 or 1e-136 < (*.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 96.1

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

   5. Applied rewrites 96.1%

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

   if -0.0050000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-136 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.005:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-136}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.0%

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

   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 70.9

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

   5. Applied rewrites 70.9%

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

   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 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.1%

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

4. Final simplification 65.4%

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

Alternative 7: 35.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.002:\\ \;\;\;\;\left(\left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.002)
   (*
    (*
     (- re -1.0)
     (fma
      (fma
       (fma (* im im) -0.0001984126984126984 0.008333333333333333)
       (* im im)
       -0.16666666666666666)
      (* im im)
      1.0))
    im)
   (fma (fma (* (fma 0.16666666666666666 re 0.5) im) re im) re im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.002) {
		tmp = ((re - -1.0) * fma(fma(fma((im * im), -0.0001984126984126984, 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0)) * im;
	} else {
		tmp = fma(fma((fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.002)
		tmp = Float64(Float64(Float64(re - -1.0) * fma(fma(fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)) * im);
	else
		tmp = fma(fma(Float64(fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(re - -1.0), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-3

   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 69.2%

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

   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 42.5%

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

   if 2e-3 < (*.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 23.1

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

   5. Applied rewrites 23.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 14.0%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.002:\\ \;\;\;\;\left(\left(re - -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im \cdot im, im\right) \cdot \left(1 + re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.002)
   (* (fma (* -0.16666666666666666 im) (* im im) im) (+ 1.0 re))
   (fma (fma (* (fma 0.16666666666666666 re 0.5) im) re im) re im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.002) {
		tmp = fma((-0.16666666666666666 * im), (im * im), im) * (1.0 + re);
	} else {
		tmp = fma(fma((fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.002)
		tmp = Float64(fma(Float64(-0.16666666666666666 * im), Float64(im * im), im) * Float64(1.0 + re));
	else
		tmp = fma(fma(Float64(fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(-0.16666666666666666 * im), $MachinePrecision] * N[(im * im), $MachinePrecision] + im), $MachinePrecision] * N[(1.0 + re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-3

   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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. cube-unmult N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, {im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, im\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{3}}, {im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, im\right) \]

      9. sub-neg N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, \color{blue}{{im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, im\right) \]

      10. *-commutative N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), im\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2} + \color{blue}{\frac{-1}{6}}, im\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, \color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}, {im}^{2}, \frac{-1}{6}\right)}, im\right) \]

      13. +-commutative N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, \mathsf{fma}\left(\color{blue}{\frac{-1}{5040} \cdot {im}^{2} + \frac{1}{120}}, {im}^{2}, \frac{-1}{6}\right), im\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{5040}, {im}^{2}, \frac{1}{120}\right)}, {im}^{2}, \frac{-1}{6}\right), im\right) \]

      15. unpow2 N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{im \cdot im}, \frac{1}{120}\right), {im}^{2}, \frac{-1}{6}\right), im\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{im \cdot im}, \frac{1}{120}\right), {im}^{2}, \frac{-1}{6}\right), im\right) \]

      17. unpow2 N/A

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, im \cdot im, \frac{1}{120}\right), \color{blue}{im \cdot im}, \frac{-1}{6}\right), im\right) \]

      18. lower-*.f64 42.5

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), \color{blue}{im \cdot im}, -0.16666666666666666\right), im\right) \]

   8. Applied rewrites 42.5%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im\right)} \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 42.0%

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

   13. Applied rewrites 42.0%

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

   if 2e-3 < (*.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 23.1

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

   5. Applied rewrites 23.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 14.0%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im \cdot im, im\right) \cdot \left(1 + re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;\sin im \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot t\_0, re, t\_0\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (fma (* im im) -0.0001984126984126984 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0)))
   (if (<= (sin im) 0.002)
     (* (fma (* (fma (fma 0.16666666666666666 re 0.5) re 1.0) t_0) re t_0) im)
     (fma (fma (* (fma 0.16666666666666666 re 0.5) im) re im) re im))))

C

double code(double re, double im) {
	double t_0 = fma(fma(fma((im * im), -0.0001984126984126984, 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double tmp;
	if (sin(im) <= 0.002) {
		tmp = fma((fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * t_0), re, t_0) * im;
	} else {
		tmp = fma(fma((fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	tmp = 0.0
	if (sin(im) <= 0.002)
		tmp = Float64(fma(Float64(fma(fma(0.16666666666666666, re, 0.5), re, 1.0) * t_0), re, t_0) * im);
	else
		tmp = fma(fma(Float64(fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Sin[im], $MachinePrecision], 0.002], N[(N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 im) < 2e-3

   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 75.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 52.8%

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

   if 2e-3 < (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 28.8

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

   5. Applied rewrites 28.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 7.1%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\\ \mathbf{if}\;\sin im \leq 5 \cdot 10^{-241}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(0.5, re, 1\right), re, t\_0\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (fma (* im im) -0.0001984126984126984 0.008333333333333333)
           (* im im)
           -0.16666666666666666)
          (* im im)
          1.0)))
   (if (<= (sin im) 5e-241)
     (* (fma (* t_0 (fma 0.5 re 1.0)) re t_0) im)
     (fma (fma (* (fma 0.16666666666666666 re 0.5) im) re im) re im))))

C

double code(double re, double im) {
	double t_0 = fma(fma(fma((im * im), -0.0001984126984126984, 0.008333333333333333), (im * im), -0.16666666666666666), (im * im), 1.0);
	double tmp;
	if (sin(im) <= 5e-241) {
		tmp = fma((t_0 * fma(0.5, re, 1.0)), re, t_0) * im;
	} else {
		tmp = fma(fma((fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(fma(Float64(im * im), -0.0001984126984126984, 0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), 1.0)
	tmp = 0.0
	if (sin(im) <= 5e-241)
		tmp = Float64(fma(Float64(t_0 * fma(0.5, re, 1.0)), re, t_0) * im);
	else
		tmp = fma(fma(Float64(fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Sin[im], $MachinePrecision], 5e-241], N[(N[(N[(t$95$0 * N[(0.5 * re + 1.0), $MachinePrecision]), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 im) < 4.9999999999999998e-241

   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 69.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.9%

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

   if 4.9999999999999998e-241 < (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 54.7

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

   5. Applied rewrites 54.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.4%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \leq 5 \cdot 10^{-241}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(0.5, re, 1\right), re, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, 1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00039:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00039)
   (* im (exp re))
   (if (<= re 6e-10)
     (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
     (if (<= re 1.05e+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 <= -0.00039) {
		tmp = im * exp(re);
	} else if (re <= 6e-10) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else if (re <= 1.05e+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 <= -0.00039)
		tmp = Float64(im * exp(re));
	elseif (re <= 6e-10)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	elseif (re <= 1.05e+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, -0.00039], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6e-10], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.05e+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 < -3.89999999999999993e-4

   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 -3.89999999999999993e-4 < re < 6e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 6e-10 < re < 1.0500000000000001e103

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 84.2%

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

   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 84.2%

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

   if 1.0500000000000001e103 < 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 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00039:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, 1\right) \cdot e^{re}\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \leq 5 \cdot 10^{-241}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666 \cdot im, im\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin im) 5e-241)
   (*
    (fma (* im im) (* -0.16666666666666666 im) im)
    (fma (fma 0.5 re 1.0) re 1.0))
   (fma (fma (* (fma 0.16666666666666666 re 0.5) im) re im) re im)))

C

double code(double re, double im) {
	double tmp;
	if (sin(im) <= 5e-241) {
		tmp = fma((im * im), (-0.16666666666666666 * im), im) * fma(fma(0.5, re, 1.0), re, 1.0);
	} else {
		tmp = fma(fma((fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(im) <= 5e-241)
		tmp = Float64(fma(Float64(im * im), Float64(-0.16666666666666666 * im), im) * fma(fma(0.5, re, 1.0), re, 1.0));
	else
		tmp = fma(fma(Float64(fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[im], $MachinePrecision], 5e-241], N[(N[(N[(im * im), $MachinePrecision] * N[(-0.16666666666666666 * im), $MachinePrecision] + im), $MachinePrecision] * N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 im) < 4.9999999999999998e-241

   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 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 44.4

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

   8. Applied rewrites 44.4%

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

   10. Applied rewrites 44.4%

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

   if 4.9999999999999998e-241 < (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 54.7

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

   5. Applied rewrites 54.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.4%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \leq 5 \cdot 10^{-241}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666 \cdot im, im\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot im, re, im\right), re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.0% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (fma (fma (* (fma 0.16666666666666666 re 0.5) im) re im) re im))

C

double code(double re, double im) {
	return fma(fma((fma(0.16666666666666666, re, 0.5) * im), re, im), re, im);
}

Julia

function code(re, im)
	return fma(fma(Float64(fma(0.16666666666666666, re, 0.5) * im), re, im), re, im)
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision] * re + im), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 66.0

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

5. Applied rewrites 66.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 38.2%

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

9. Final simplification 38.2%

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

Alternative 14: 34.5% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 66.0

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

5. Applied rewrites 66.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 36.0%

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

Alternative 15: 29.7% accurate, 29.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 66.0

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

5. Applied rewrites 66.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 32.9%

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

Alternative 16: 6.9% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 66.0

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

5. Applied rewrites 66.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 32.9%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 7.3%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))