math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 98.5%

Time: 16.3s

Alternatives: 17

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: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ t_1 := e^{re} \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im\_m, -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right), im\_m\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\_m \cdot \left(re + 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\_m \cdot \left(re + \mathsf{fma}\left(re, re \cdot 0.5, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))) (t_1 (* (exp re) im_m)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (+ re 1.0) (fma im_m (* -0.16666666666666666 (* im_m im_m)) im_m))
      (if (<= t_0 -0.02)
        (* (sin im_m) (+ re 1.0))
        (if (<= t_0 5e-21)
          t_1
          (if (<= t_0 1.0)
            (* (sin im_m) (+ re (fma re (* re 0.5) 1.0)))
            t_1)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double t_1 = exp(re) * im_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re + 1.0) * fma(im_m, (-0.16666666666666666 * (im_m * im_m)), im_m);
	} else if (t_0 <= -0.02) {
		tmp = sin(im_m) * (re + 1.0);
	} else if (t_0 <= 5e-21) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im_m) * (re + fma(re, (re * 0.5), 1.0));
	} else {
		tmp = t_1;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(re) * sin(im_m))
	t_1 = Float64(exp(re) * im_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re + 1.0) * fma(im_m, Float64(-0.16666666666666666 * Float64(im_m * im_m)), im_m));
	elseif (t_0 <= -0.02)
		tmp = Float64(sin(im_m) * Float64(re + 1.0));
	elseif (t_0 <= 5e-21)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(im_m) * Float64(re + fma(re, Float64(re * 0.5), 1.0)));
	else
		tmp = t_1;
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(re + 1.0), $MachinePrecision] * N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[Sin[im$95$m], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-21], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[Sin[im$95$m], $MachinePrecision] * N[(re + N[(re * N[(re * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.1

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

   8. Applied rewrites 28.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999973e-21 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if 4.99999999999999973e-21 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-21}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + \mathsf{fma}\left(re, re \cdot 0.5, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ t_1 := e^{re} \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im\_m, -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right), im\_m\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\_m \cdot \left(re + 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\_m \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))) (t_1 (* (exp re) im_m)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (+ re 1.0) (fma im_m (* -0.16666666666666666 (* im_m im_m)) im_m))
      (if (<= t_0 -0.02)
        (* (sin im_m) (+ re 1.0))
        (if (<= t_0 5e-21)
          t_1
          (if (<= t_0 1.0)
            (* (sin im_m) (fma re (fma re 0.5 1.0) 1.0))
            t_1)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double t_1 = exp(re) * im_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re + 1.0) * fma(im_m, (-0.16666666666666666 * (im_m * im_m)), im_m);
	} else if (t_0 <= -0.02) {
		tmp = sin(im_m) * (re + 1.0);
	} else if (t_0 <= 5e-21) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im_m) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else {
		tmp = t_1;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(re) * sin(im_m))
	t_1 = Float64(exp(re) * im_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re + 1.0) * fma(im_m, Float64(-0.16666666666666666 * Float64(im_m * im_m)), im_m));
	elseif (t_0 <= -0.02)
		tmp = Float64(sin(im_m) * Float64(re + 1.0));
	elseif (t_0 <= 5e-21)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(im_m) * fma(re, fma(re, 0.5, 1.0), 1.0));
	else
		tmp = t_1;
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(re + 1.0), $MachinePrecision] * N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[Sin[im$95$m], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-21], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[Sin[im$95$m], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.1

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

   8. Applied rewrites 28.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999973e-21 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if 4.99999999999999973e-21 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-21}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ t_1 := \sin im\_m \cdot \left(re + 1\right)\\ t_2 := e^{re} \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im\_m, -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right), im\_m\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m)))
        (t_1 (* (sin im_m) (+ re 1.0)))
        (t_2 (* (exp re) im_m)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (+ re 1.0) (fma im_m (* -0.16666666666666666 (* im_m im_m)) im_m))
      (if (<= t_0 -0.02)
        t_1
        (if (<= t_0 5e-21) t_2 (if (<= t_0 1.0) t_1 t_2)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double t_1 = sin(im_m) * (re + 1.0);
	double t_2 = exp(re) * im_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re + 1.0) * fma(im_m, (-0.16666666666666666 * (im_m * im_m)), im_m);
	} else if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 5e-21) {
		tmp = t_2;
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(re) * sin(im_m))
	t_1 = Float64(sin(im_m) * Float64(re + 1.0))
	t_2 = Float64(exp(re) * im_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re + 1.0) * fma(im_m, Float64(-0.16666666666666666 * Float64(im_m * im_m)), im_m));
	elseif (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 5e-21)
		tmp = t_2;
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im$95$m], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(re + 1.0), $MachinePrecision] * N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], t$95$1, If[LessEqual[t$95$0, 5e-21], t$95$2, If[LessEqual[t$95$0, 1.0], t$95$1, t$95$2]]]]), $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 re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.1

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

   8. Applied rewrites 28.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 4.99999999999999973e-21 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999973e-21 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-21}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ t_1 := e^{re} \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im\_m, -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right), im\_m\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))) (t_1 (* (exp re) im_m)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (+ re 1.0) (fma im_m (* -0.16666666666666666 (* im_m im_m)) im_m))
      (if (<= t_0 -0.02)
        (sin im_m)
        (if (<= t_0 5e-21) t_1 (if (<= t_0 1.0) (sin im_m) t_1)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double t_1 = exp(re) * im_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re + 1.0) * fma(im_m, (-0.16666666666666666 * (im_m * im_m)), im_m);
	} else if (t_0 <= -0.02) {
		tmp = sin(im_m);
	} else if (t_0 <= 5e-21) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im_m);
	} else {
		tmp = t_1;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(re) * sin(im_m))
	t_1 = Float64(exp(re) * im_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re + 1.0) * fma(im_m, Float64(-0.16666666666666666 * Float64(im_m * im_m)), im_m));
	elseif (t_0 <= -0.02)
		tmp = sin(im_m);
	elseif (t_0 <= 5e-21)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im_m);
	else
		tmp = t_1;
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(re + 1.0), $MachinePrecision] * N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[im$95$m], $MachinePrecision], If[LessEqual[t$95$0, 5e-21], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im$95$m], $MachinePrecision], t$95$1]]]]), $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 re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.1

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

   8. Applied rewrites 28.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 4.99999999999999973e-21 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999973e-21 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im, -0.16666666666666666 \cdot \left(im \cdot im\right), im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-21}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\\ t_1 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(re + 1\right) \cdot \mathsf{fma}\left(im\_m, t\_0, im\_m\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;\sin im\_m\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;im\_m \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.008333333333333333, -0.16666666666666666\right), im\_m \cdot \left(im\_m \cdot im\_m\right), im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* im_m im_m)))
        (t_1 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* (+ re 1.0) (fma im_m t_0 im_m))
      (if (<= t_1 -0.02)
        (sin im_m)
        (if (<= t_1 0.0)
          (* im_m t_0)
          (if (<= t_1 1.0)
            (sin im_m)
            (*
             (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
             (fma
              (fma (* im_m im_m) 0.008333333333333333 -0.16666666666666666)
              (* im_m (* im_m im_m))
              im_m)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -0.16666666666666666 * (im_m * im_m);
	double t_1 = exp(re) * sin(im_m);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (re + 1.0) * fma(im_m, t_0, im_m);
	} else if (t_1 <= -0.02) {
		tmp = sin(im_m);
	} else if (t_1 <= 0.0) {
		tmp = im_m * t_0;
	} else if (t_1 <= 1.0) {
		tmp = sin(im_m);
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(fma((im_m * im_m), 0.008333333333333333, -0.16666666666666666), (im_m * (im_m * im_m)), im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(-0.16666666666666666 * Float64(im_m * im_m))
	t_1 = Float64(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(re + 1.0) * fma(im_m, t_0, im_m));
	elseif (t_1 <= -0.02)
		tmp = sin(im_m);
	elseif (t_1 <= 0.0)
		tmp = Float64(im_m * t_0);
	elseif (t_1 <= 1.0)
		tmp = sin(im_m);
	else
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(fma(Float64(im_m * im_m), 0.008333333333333333, -0.16666666666666666), Float64(im_m * Float64(im_m * im_m)), im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(re + 1.0), $MachinePrecision] * N[(im$95$m * t$95$0 + im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], N[Sin[im$95$m], $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(im$95$m * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[im$95$m], $MachinePrecision], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 28.1

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

   8. Applied rewrites 28.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or -0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 36.4%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{6} \cdot {im}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 32.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.9

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 21.2

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

   8. Applied rewrites 21.2%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 53.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 53.4% accurate, 0.8× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.0)
    (* im_m (* -0.16666666666666666 (* im_m im_m)))
    (*
     (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
     (fma
      (fma (* im_m im_m) 0.008333333333333333 -0.16666666666666666)
      (* im_m (* im_m im_m))
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.0) {
		tmp = im_m * (-0.16666666666666666 * (im_m * im_m));
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(fma((im_m * im_m), 0.008333333333333333, -0.16666666666666666), (im_m * (im_m * im_m)), im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.0)
		tmp = Float64(im_m * Float64(-0.16666666666666666 * Float64(im_m * im_m)));
	else
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * fma(fma(Float64(im_m * im_m), 0.008333333333333333, -0.16666666666666666), Float64(im_m * Float64(im_m * im_m)), im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 46.3

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

   5. Applied rewrites 46.3%

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

   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 25.5%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{6} \cdot {im}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 66.4

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 41.0

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

   8. Applied rewrites 41.0%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 52.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 51.5% accurate, 0.8× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.0)
    (* im_m (* -0.16666666666666666 (* im_m im_m)))
    (*
     (fma re (fma re 0.5 1.0) 1.0)
     (fma
      (fma (* im_m im_m) 0.008333333333333333 -0.16666666666666666)
      (* im_m (* im_m im_m))
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.0) {
		tmp = im_m * (-0.16666666666666666 * (im_m * im_m));
	} else {
		tmp = fma(re, fma(re, 0.5, 1.0), 1.0) * fma(fma((im_m * im_m), 0.008333333333333333, -0.16666666666666666), (im_m * (im_m * im_m)), im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.0)
		tmp = Float64(im_m * Float64(-0.16666666666666666 * Float64(im_m * im_m)));
	else
		tmp = Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(fma(Float64(im_m * im_m), 0.008333333333333333, -0.16666666666666666), Float64(im_m * Float64(im_m * im_m)), im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 46.3

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

   5. Applied rewrites 46.3%

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

   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 25.5%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{6} \cdot {im}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 86.6

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 52.3

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

   8. Applied rewrites 52.3%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), im \cdot \left(im \cdot im\right), im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 52.9% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.0)
    (* im_m (* -0.16666666666666666 (* im_m im_m)))
    (* im_m (+ re (+ 1.0 (* re (* re (fma re 0.16666666666666666 0.5)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.0) {
		tmp = im_m * (-0.16666666666666666 * (im_m * im_m));
	} else {
		tmp = im_m * (re + (1.0 + (re * (re * fma(re, 0.16666666666666666, 0.5)))));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.0)
		tmp = Float64(im_m * Float64(-0.16666666666666666 * Float64(im_m * im_m)));
	else
		tmp = Float64(im_m * Float64(re + Float64(1.0 + Float64(re * Float64(re * fma(re, 0.16666666666666666, 0.5))))));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re + N[(1.0 + N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 46.3

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

   5. Applied rewrites 46.3%

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

   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 25.5%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{6} \cdot {im}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 51.1%

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

   10. Applied rewrites 51.1%

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

4. Final simplification 34.5%

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

Alternative 9: 52.9% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.0)
    (* im_m (* -0.16666666666666666 (* im_m im_m)))
    (* im_m (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.0) {
		tmp = im_m * (-0.16666666666666666 * (im_m * im_m));
	} else {
		tmp = im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.0)
		tmp = Float64(im_m * Float64(-0.16666666666666666 * Float64(im_m * im_m)));
	else
		tmp = Float64(im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 46.3

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

   5. Applied rewrites 46.3%

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

   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 25.5%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{6} \cdot {im}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 51.1%

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

Alternative 10: 51.5% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.0)
    (* im_m (* -0.16666666666666666 (* im_m im_m)))
    (fma re (fma im_m (* re (fma re 0.16666666666666666 0.5)) im_m) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.0) {
		tmp = im_m * (-0.16666666666666666 * (im_m * im_m));
	} else {
		tmp = fma(re, fma(im_m, (re * fma(re, 0.16666666666666666, 0.5)), im_m), im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.0)
		tmp = Float64(im_m * Float64(-0.16666666666666666 * Float64(im_m * im_m)));
	else
		tmp = fma(re, fma(im_m, Float64(re * fma(re, 0.16666666666666666, 0.5)), im_m), im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision] + im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 46.3

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

   5. Applied rewrites 46.3%

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

   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 25.5%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{6} \cdot {im}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 56.9

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

   5. Applied rewrites 56.9%

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

   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 51.1%

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

Alternative 11: 50.5% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.0)
    (* im_m (* -0.16666666666666666 (* im_m im_m)))
    (* im_m (fma re (fma re 0.5 1.0) 1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.0) {
		tmp = im_m * (-0.16666666666666666 * (im_m * im_m));
	} else {
		tmp = im_m * fma(re, fma(re, 0.5, 1.0), 1.0);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.0)
		tmp = Float64(im_m * Float64(-0.16666666666666666 * Float64(im_m * im_m)));
	else
		tmp = Float64(im_m * fma(re, fma(re, 0.5, 1.0), 1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 46.3

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

   5. Applied rewrites 46.3%

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

   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 25.5%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{6} \cdot {im}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 50.9%

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

Alternative 12: 47.6% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.0)
    (* im_m (* -0.16666666666666666 (* im_m im_m)))
    (fma re (fma re (* im_m 0.5) im_m) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.0) {
		tmp = im_m * (-0.16666666666666666 * (im_m * im_m));
	} else {
		tmp = fma(re, fma(re, (im_m * 0.5), im_m), im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.0)
		tmp = Float64(im_m * Float64(-0.16666666666666666 * Float64(im_m * im_m)));
	else
		tmp = fma(re, fma(re, Float64(im_m * 0.5), im_m), im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(im$95$m * 0.5), $MachinePrecision] + im$95$m), $MachinePrecision] + im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 46.3

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

   5. Applied rewrites 46.3%

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

   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 25.5%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{6} \cdot {im}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.8%

      \[\leadsto im \cdot 1 \]

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 45.5%

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

4. Final simplification 32.2%

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

Alternative 13: 42.8% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.0)
    (* im_m (* -0.16666666666666666 (* im_m im_m)))
    (fma im_m re im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.0) {
		tmp = im_m * (-0.16666666666666666 * (im_m * im_m));
	} else {
		tmp = fma(im_m, re, im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.0)
		tmp = Float64(im_m * Float64(-0.16666666666666666 * Float64(im_m * im_m)));
	else
		tmp = fma(im_m, re, im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * re + im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 46.3

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

   5. Applied rewrites 46.3%

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

   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 25.5%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{6} \cdot {im}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.9%

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

Alternative 14: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(e^{re} \cdot \sin im\_m\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* (exp re) (sin im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (exp(re) * sin(im_m));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (exp(re) * sin(im_m))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (Math.exp(re) * Math.sin(im_m));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (math.exp(re) * math.sin(im_m))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(exp(re) * sin(im_m)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (exp(re) * sin(im_m));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

Alternative 15: 30.1% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.45 \cdot 10^{+39}:\\ \;\;\;\;im\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 2.45e+39) (* im_m 1.0) (* re im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.45e+39) {
		tmp = im_m * 1.0;
	} else {
		tmp = re * im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.45d+39) then
        tmp = im_m * 1.0d0
    else
        tmp = re * im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.45e+39) {
		tmp = im_m * 1.0;
	} else {
		tmp = re * im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.45e+39:
		tmp = im_m * 1.0
	else:
		tmp = re * im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.45e+39)
		tmp = Float64(im_m * 1.0);
	else
		tmp = Float64(re * im_m);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.45e+39)
		tmp = im_m * 1.0;
	else
		tmp = re * im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.45e+39], N[(im$95$m * 1.0), $MachinePrecision], N[(re * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.44999999999999994e39

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.1%

      \[\leadsto im \cdot 1 \]

   if 2.44999999999999994e39 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 24.3

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

   5. Applied rewrites 24.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 2.5%

      \[\leadsto im \cdot 1 \]

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 10.2%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 11.4%

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

Alternative 16: 30.1% accurate, 29.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \mathsf{fma}\left(im\_m, re, im\_m\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (fma im_m re im_m)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * fma(im_m, re, im_m);
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * fma(im_m, re, im_m))
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * re + im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-exp.f64 64.8

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

5. Applied rewrites 64.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 29.0%

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

Alternative 17: 6.8% accurate, 34.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(re \cdot im\_m\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* re im_m)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (re * im_m);
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (re * im_m)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (re * im_m);
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (re * im_m)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(re * im_m))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (re * im_m);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(re * im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-exp.f64 64.8

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

5. Applied rewrites 64.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 26.5%

   \[\leadsto im \cdot 1 \]

9. Taylor expanded in re around 0

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

11. Applied rewrites 29.0%

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

12. Taylor expanded in re around inf

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

14. Applied rewrites 6.3%

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

Reproduce

herbie shell --seed 2024238 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))