math.sin on complex, real part

Percentage Accurate: 100.0% → 99.4%

Time: 4.6s

Alternatives: 19

Speedup: 1.0×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := 2 \cdot \cosh im\\ t_1 := 0.5 \cdot \sin re\_m\\ t_2 := t\_1 \cdot \left(e^{-im} + e^{im}\right)\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(re\_m \cdot re\_m\right) \cdot -0.08333333333333333\right)\right) \cdot re\_m\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (let* ((t_0 (* 2.0 (cosh im)))
        (t_1 (* 0.5 (sin re_m)))
        (t_2 (* t_1 (+ (exp (- im)) (exp im)))))
   (*
    re_s
    (if (<= t_2 (- INFINITY))
      (* (* t_0 (* (* re_m re_m) -0.08333333333333333)) re_m)
      (if (<= t_2 1.0) (* t_1 (fma im im 2.0)) (* (* re_m 0.5) t_0))))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double t_0 = 2.0 * cosh(im);
	double t_1 = 0.5 * sin(re_m);
	double t_2 = t_1 * (exp(-im) + exp(im));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 * ((re_m * re_m) * -0.08333333333333333)) * re_m;
	} else if (t_2 <= 1.0) {
		tmp = t_1 * fma(im, im, 2.0);
	} else {
		tmp = (re_m * 0.5) * t_0;
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	t_0 = Float64(2.0 * cosh(im))
	t_1 = Float64(0.5 * sin(re_m))
	t_2 = Float64(t_1 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(Float64(re_m * re_m) * -0.08333333333333333)) * re_m);
	elseif (t_2 <= 1.0)
		tmp = Float64(t_1 * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(re_m * 0.5) * t_0);
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := Block[{t$95$0 = N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(re$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(t$95$0 * N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * re$95$m), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[(t$95$1 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

   5. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   6. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot \left({re}^{2} \cdot \frac{-1}{12}\right)\right) \cdot re \]

      3. pow2 N/A

         \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right) \cdot re \]

      4. lift-*.f64 23.4

         \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

   8. Applied rewrites 23.4%

      \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 77.1

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   5. Applied rewrites 77.1%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\_m\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right)\right) \cdot re\_m\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re_m))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (*
    re_s
    (if (<= t_1 (- INFINITY))
      (*
       (*
        (fma
         (fma
          (fma 0.002777777777777778 (* im im) 0.08333333333333333)
          (* im im)
          1.0)
         (* im im)
         2.0)
        (fma (* re_m re_m) -0.08333333333333333 0.5))
       re_m)
      (if (<= t_1 1.0)
        (* t_0 (fma im im 2.0))
        (* (* re_m 0.5) (* 2.0 (cosh im))))))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double t_0 = 0.5 * sin(re_m);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * fma((re_m * re_m), -0.08333333333333333, 0.5)) * re_m;
	} else if (t_1 <= 1.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = (re_m * 0.5) * (2.0 * cosh(im));
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	t_0 = Float64(0.5 * sin(re_m))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * fma(Float64(re_m * re_m), -0.08333333333333333, 0.5)) * re_m);
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(re_m * 0.5) * Float64(2.0 * cosh(im)));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(re$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

   5. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]
   7. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      2. sub0-neg N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      4. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      14. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      16. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      17. lower-*.f64 65.4

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Applied rewrites 65.4%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 77.1

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   5. Applied rewrites 77.1%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right)\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right)\right) \cdot re\_m\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im)))))
   (*
    re_s
    (if (<= t_0 (- INFINITY))
      (*
       (*
        (fma
         (fma
          (fma 0.002777777777777778 (* im im) 0.08333333333333333)
          (* im im)
          1.0)
         (* im im)
         2.0)
        (fma (* re_m re_m) -0.08333333333333333 0.5))
       re_m)
      (if (<= t_0 1.0) (sin re_m) (* (* re_m 0.5) (* 2.0 (cosh im))))))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double t_0 = (0.5 * sin(re_m)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * fma((re_m * re_m), -0.08333333333333333, 0.5)) * re_m;
	} else if (t_0 <= 1.0) {
		tmp = sin(re_m);
	} else {
		tmp = (re_m * 0.5) * (2.0 * cosh(im));
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	t_0 = Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * fma(Float64(re_m * re_m), -0.08333333333333333, 0.5)) * re_m);
	elseif (t_0 <= 1.0)
		tmp = sin(re_m);
	else
		tmp = Float64(Float64(re_m * 0.5) * Float64(2.0 * cosh(im)));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(re$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re$95$m], $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

   5. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]
   7. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      2. sub0-neg N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      4. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      14. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      16. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      17. lower-*.f64 65.4

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Applied rewrites 65.4%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 99.3

         \[\leadsto \sin re \]

   5. Applied rewrites 99.3%

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

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 77.1

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   5. Applied rewrites 77.1%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right)\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right)\right) \cdot re\_m\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im)))))
   (*
    re_s
    (if (<= t_0 (- INFINITY))
      (*
       (*
        (fma
         (fma
          (fma 0.002777777777777778 (* im im) 0.08333333333333333)
          (* im im)
          1.0)
         (* im im)
         2.0)
        (fma (* re_m re_m) -0.08333333333333333 0.5))
       re_m)
      (if (<= t_0 1.0)
        (sin re_m)
        (*
         (* re_m 0.5)
         (fma
          (*
           (fma
            (* (fma (* 0.002777777777777778 im) im 0.08333333333333333) im)
            im
            1.0)
           im)
          im
          2.0)))))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double t_0 = (0.5 * sin(re_m)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * fma((re_m * re_m), -0.08333333333333333, 0.5)) * re_m;
	} else if (t_0 <= 1.0) {
		tmp = sin(re_m);
	} else {
		tmp = (re_m * 0.5) * fma((fma((fma((0.002777777777777778 * im), im, 0.08333333333333333) * im), im, 1.0) * im), im, 2.0);
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	t_0 = Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * fma(Float64(re_m * re_m), -0.08333333333333333, 0.5)) * re_m);
	elseif (t_0 <= 1.0)
		tmp = sin(re_m);
	else
		tmp = Float64(Float64(re_m * 0.5) * fma(Float64(fma(Float64(fma(Float64(0.002777777777777778 * im), im, 0.08333333333333333) * im), im, 1.0) * im), im, 2.0));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(re$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re$95$m), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re$95$m], $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(0.002777777777777778 * im), $MachinePrecision] * im + 0.08333333333333333), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * im), $MachinePrecision] * im + 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

   5. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]
   7. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      2. sub0-neg N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      4. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      14. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      16. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      17. lower-*.f64 65.4

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Applied rewrites 65.4%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 99.3

         \[\leadsto \sin re \]

   5. Applied rewrites 99.3%

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

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 77.1

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   5. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      2. sub0-neg N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{\color{blue}{2}}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \]

      8. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. +-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      12. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      14. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      16. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      17. lower-*.f64 67.5

          \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 67.5%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im\right) \cdot im + 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im, im, 2\right) \]

   10. Applied rewrites 67.5%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, im, 2\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot im, 1\right) \cdot im, im, 2\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot im, 1\right) \cdot im, im, 2\right) \]

       3. lift-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{360} + \frac{1}{12}, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       4. *-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       5. associate-*l* N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im, im, 2\right) \]

       7. associate-*r* N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot im\right) \cdot im + 1\right) \cdot im, im, 2\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

       10. lift-fma.f64 N/A

           \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot im, im, \frac{1}{12}\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

       11. lift-*.f64 67.5

           \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

   12. Applied rewrites 67.5%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.08333333333333333, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right)\right) \cdot re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (*
  re_s
  (if (<= (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im))) 0.0002)
    (*
     (*
      (fma (fma im (* im 0.08333333333333333) 1.0) (* im im) 2.0)
      (fma (* re_m re_m) -0.08333333333333333 0.5))
     re_m)
    (*
     (* re_m 0.5)
     (fma
      (*
       (fma
        (* (fma (* 0.002777777777777778 im) im 0.08333333333333333) im)
        im
        1.0)
       im)
      im
      2.0)))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double tmp;
	if (((0.5 * sin(re_m)) * (exp(-im) + exp(im))) <= 0.0002) {
		tmp = (fma(fma(im, (im * 0.08333333333333333), 1.0), (im * im), 2.0) * fma((re_m * re_m), -0.08333333333333333, 0.5)) * re_m;
	} else {
		tmp = (re_m * 0.5) * fma((fma((fma((0.002777777777777778 * im), im, 0.08333333333333333) * im), im, 1.0) * im), im, 2.0);
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0002)
		tmp = Float64(Float64(fma(fma(im, Float64(im * 0.08333333333333333), 1.0), Float64(im * im), 2.0) * fma(Float64(re_m * re_m), -0.08333333333333333, 0.5)) * re_m);
	else
		tmp = Float64(Float64(re_m * 0.5) * fma(Float64(fma(Float64(fma(Float64(0.002777777777777778 * im), im, 0.08333333333333333) * im), im, 1.0) * im), im, 2.0));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := N[(re$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(im * N[(im * 0.08333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re$95$m), $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(0.002777777777777778 * im), $MachinePrecision] * im + 0.08333333333333333), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * im), $MachinePrecision] * im + 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

   5. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]
   7. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      2. sub0-neg N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      4. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + \frac{1}{12} \cdot {im}^{2}, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{12} \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \frac{1}{12} + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left({im}^{2}, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      10. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      13. lower-*.f64 62.2

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Applied rewrites 62.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{12} + 1, im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      3. associate-*l* N/A

         \[\leadsto \left(\mathsf{fma}\left(im \cdot \left(im \cdot \frac{1}{12}\right) + 1, im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot \frac{1}{12}, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      5. lower-*.f64 62.2

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.08333333333333333, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   10. Applied rewrites 62.2%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.08333333333333333, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 54.8

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      2. sub0-neg N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{\color{blue}{2}}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \]

      8. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. +-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      12. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      14. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      16. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      17. lower-*.f64 48.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 48.0%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im\right) \cdot im + 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im, im, 2\right) \]

   10. Applied rewrites 48.0%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, im, 2\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot im, 1\right) \cdot im, im, 2\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot im, 1\right) \cdot im, im, 2\right) \]

       3. lift-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{360} + \frac{1}{12}, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       4. *-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       5. associate-*l* N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im, im, 2\right) \]

       7. associate-*r* N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot im\right) \cdot im + 1\right) \cdot im, im, 2\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

       10. lift-fma.f64 N/A

           \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot im, im, \frac{1}{12}\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

       11. lift-*.f64 48.0

           \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

   12. Applied rewrites 48.0%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot 0.08333333333333333, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (*
  re_s
  (if (<= (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im))) 0.0002)
    (* (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m) (fma im im 2.0))
    (*
     (* re_m 0.5)
     (fma
      (*
       (fma
        (* (fma (* 0.002777777777777778 im) im 0.08333333333333333) im)
        im
        1.0)
       im)
      im
      2.0)))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double tmp;
	if (((0.5 * sin(re_m)) * (exp(-im) + exp(im))) <= 0.0002) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0);
	} else {
		tmp = (re_m * 0.5) * fma((fma((fma((0.002777777777777778 * im), im, 0.08333333333333333) * im), im, 1.0) * im), im, 2.0);
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0002)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(re_m * 0.5) * fma(Float64(fma(Float64(fma(Float64(0.002777777777777778 * im), im, 0.08333333333333333) * im), im, 1.0) * im), im, 2.0));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := N[(re$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(0.002777777777777778 * im), $MachinePrecision] * im + 0.08333333333333333), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * im), $MachinePrecision] * im + 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.8

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 74.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. pow2 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lift-*.f64 55.4

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 54.8

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      2. sub0-neg N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{\color{blue}{2}}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \]

      8. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. +-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      12. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      14. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      16. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      17. lower-*.f64 48.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 48.0%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im\right) \cdot im + 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im, im, 2\right) \]

   10. Applied rewrites 48.0%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, im, 2\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot im, 1\right) \cdot im, im, 2\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{360}, \frac{1}{12}\right), im \cdot im, 1\right) \cdot im, im, 2\right) \]

       3. lift-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{360} + \frac{1}{12}, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       4. *-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       5. associate-*l* N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im, im, 2\right) \]

       7. associate-*r* N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot im\right) \cdot im + 1\right) \cdot im, im, 2\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot im\right) \cdot im + \frac{1}{12}\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

       10. lift-fma.f64 N/A

           \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot im, im, \frac{1}{12}\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

       11. lift-*.f64 48.0

           \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]

   12. Applied rewrites 48.0%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot im, im, 0.08333333333333333\right) \cdot im, im, 1\right) \cdot im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(0.002777777777777778 \cdot im\right) \cdot im, im \cdot im, 1\right) \cdot im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (*
  re_s
  (if (<= (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im))) 0.0002)
    (* (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m) (fma im im 2.0))
    (*
     (* re_m 0.5)
     (fma
      (* (fma (* (* 0.002777777777777778 im) im) (* im im) 1.0) im)
      im
      2.0)))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double tmp;
	if (((0.5 * sin(re_m)) * (exp(-im) + exp(im))) <= 0.0002) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0);
	} else {
		tmp = (re_m * 0.5) * fma((fma(((0.002777777777777778 * im) * im), (im * im), 1.0) * im), im, 2.0);
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0002)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(re_m * 0.5) * fma(Float64(fma(Float64(Float64(0.002777777777777778 * im) * im), Float64(im * im), 1.0) * im), im, 2.0));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := N[(re$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(0.002777777777777778 * im), $MachinePrecision] * im), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision] * im + 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.8

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 74.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. pow2 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lift-*.f64 55.4

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 54.8

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      2. sub0-neg N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{\color{blue}{2}}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \]

      8. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. +-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      12. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      14. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      16. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      17. lower-*.f64 48.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 48.0%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im\right) \cdot im + 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(im \cdot im\right) + \frac{1}{12}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im, im, 2\right) \]

   10. Applied rewrites 48.0%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, im, 2\right) \]

   11. Taylor expanded in im around inf

       \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2}, im \cdot im, 1\right) \cdot im, im, 2\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2}, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       2. associate-*l* N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2}, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       3. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot im, im, 2\right) \]

       4. associate-*l* N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{360} \cdot im\right) \cdot im, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{360} \cdot im\right) \cdot im, im \cdot im, 1\right) \cdot im, im, 2\right) \]

       6. lift-*.f64 48.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(0.002777777777777778 \cdot im\right) \cdot im, im \cdot im, 1\right) \cdot im, im, 2\right) \]

   13. Applied rewrites 48.0%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(0.002777777777777778 \cdot im\right) \cdot im, im \cdot im, 1\right) \cdot im, im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(0.002777777777777778 \cdot im\right) \cdot im, im \cdot im, 1\right) \cdot im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right) \cdot im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (*
  re_s
  (if (<= (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im))) 0.0002)
    (* (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m) (fma im im 2.0))
    (*
     (* re_m 0.5)
     (fma (* (fma (* im im) 0.08333333333333333 1.0) im) im 2.0)))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double tmp;
	if (((0.5 * sin(re_m)) * (exp(-im) + exp(im))) <= 0.0002) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0);
	} else {
		tmp = (re_m * 0.5) * fma((fma((im * im), 0.08333333333333333, 1.0) * im), im, 2.0);
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0002)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(re_m * 0.5) * fma(Float64(fma(Float64(im * im), 0.08333333333333333, 1.0) * im), im, 2.0));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := N[(re$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * im), $MachinePrecision] * im + 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.8

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 74.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. pow2 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lift-*.f64 55.4

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 54.8

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)} \]

   6. Taylor expanded in im around 0

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

      1. cosh-undef-rev N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \]

      2. sub0-neg N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(1 + \frac{1}{12} \cdot {im}^{2}, {im}^{\color{blue}{2}}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{12} \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      8. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \frac{1}{12} + 1, {im}^{2}, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({im}^{2}, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \]

      10. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \]

      12. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      13. lower-*.f64 45.2

          \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 45.2%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), \color{blue}{im \cdot im}, 2\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{1}{12} + 1\right) \cdot \left(im \cdot im\right) + 2\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{1}{12} + 1\right) \cdot im\right) \cdot im + 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot \frac{1}{12} + 1\right) \cdot im, im, 2\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot \frac{1}{12} + 1\right) \cdot im, im, 2\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right) \cdot im, im, 2\right) \]

      9. lift-*.f64 45.2

         \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right) \cdot im, im, 2\right) \]

   10. Applied rewrites 45.2%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right) \cdot im, im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right) \cdot im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (*
  re_s
  (if (<= (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im))) 0.0002)
    (* (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m) (fma im im 2.0))
    (* (* re_m 0.5) (fma (* (* im im) 0.08333333333333333) (* im im) 2.0)))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double tmp;
	if (((0.5 * sin(re_m)) * (exp(-im) + exp(im))) <= 0.0002) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0);
	} else {
		tmp = (re_m * 0.5) * fma(((im * im) * 0.08333333333333333), (im * im), 2.0);
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0002)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(re_m * 0.5) * fma(Float64(Float64(im * im) * 0.08333333333333333), Float64(im * im), 2.0));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := N[(re$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.8

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 74.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. pow2 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lift-*.f64 55.4

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 54.8

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)} \]

   6. Taylor expanded in im around 0

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

      1. cosh-undef-rev N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \]

      2. sub0-neg N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(1 + \frac{1}{12} \cdot {im}^{2}, {im}^{\color{blue}{2}}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{12} \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      8. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \frac{1}{12} + 1, {im}^{2}, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({im}^{2}, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \]

      10. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \]

      12. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      13. lower-*.f64 45.2

          \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 45.2%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{12} \cdot {im}^{2}, im \cdot im, 2\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \frac{1}{12}, im \cdot im, 2\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \frac{1}{12}, im \cdot im, 2\right) \]

       3. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{12}, im \cdot im, 2\right) \]

       4. lift-*.f64 45.2

          \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right) \]

   11. Applied rewrites 45.2%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (*
  re_s
  (if (<= (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im))) 0.0002)
    (* (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m) (fma im im 2.0))
    (* (* 0.5 re_m) (fma im im 2.0)))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double tmp;
	if (((0.5 * sin(re_m)) * (exp(-im) + exp(im))) <= 0.0002) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0);
	} else {
		tmp = (0.5 * re_m) * fma(im, im, 2.0);
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0002)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(0.5 * re_m) * fma(im, im, 2.0));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := N[(re$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.8

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 74.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. pow2 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lift-*.f64 55.4

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.4

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 71.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.0%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (*
  re_s
  (if (<= (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im))) -0.05)
    (* (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m) (* im im))
    (* (* 0.5 re_m) (fma im im 2.0)))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double tmp;
	if (((0.5 * sin(re_m)) * (exp(-im) + exp(im))) <= -0.05) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * (im * im);
	} else {
		tmp = (0.5 * re_m) * fma(im, im, 2.0);
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.05)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * Float64(im * im));
	else
		tmp = Float64(Float64(0.5 * re_m) * fma(im, im, 2.0));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := N[(re$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 63.3

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 63.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. pow2 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lift-*.f64 35.0

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in im around inf

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

       1. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(im \cdot im\right) \]

       2. lift-*.f64 35.0

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(im \cdot im\right) \]

   11. Applied rewrites 35.0%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(im \cdot \color{blue}{im}\right) \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 80.8

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 80.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.8%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re\_m \cdot re\_m, 1\right) \cdot re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (*
  re_s
  (if (<= (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im))) 0.0002)
    (* (fma -0.16666666666666666 (* re_m re_m) 1.0) re_m)
    (* (* 0.5 re_m) (fma im im 2.0)))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double tmp;
	if (((0.5 * sin(re_m)) * (exp(-im) + exp(im))) <= 0.0002) {
		tmp = fma(-0.16666666666666666, (re_m * re_m), 1.0) * re_m;
	} else {
		tmp = (0.5 * re_m) * fma(im, im, 2.0);
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.0002)
		tmp = Float64(fma(-0.16666666666666666, Float64(re_m * re_m), 1.0) * re_m);
	else
		tmp = Float64(Float64(0.5 * re_m) * fma(im, im, 2.0));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := N[(re$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(-0.16666666666666666 * N[(re$95$m * re$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * re$95$m), $MachinePrecision], N[(N[(0.5 * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 51.8

         \[\leadsto \sin re \]

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 39.1

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

   8. Applied rewrites 39.1%

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

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.4

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 71.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.0%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.9% accurate, 0.9× speedup

Math

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

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (*
  re_s
  (if (<= (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im))) -0.002)
    (* (* (* re_m re_m) -0.16666666666666666) re_m)
    re_m)))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double tmp;
	if (((0.5 * sin(re_m)) * (exp(-im) + exp(im))) <= -0.002) {
		tmp = ((re_m * re_m) * -0.16666666666666666) * re_m;
	} else {
		tmp = re_m;
	}
	return re_s * tmp;
}

Fortran

re\_m =     private
re\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re_s, re_m, im)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((0.5d0 * sin(re_m)) * (exp(-im) + exp(im))) <= (-0.002d0)) then
        tmp = ((re_m * re_m) * (-0.16666666666666666d0)) * re_m
    else
        tmp = re_m
    end if
    code = re_s * tmp
end function

Java

re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
public static double code(double re_s, double re_m, double im) {
	double tmp;
	if (((0.5 * Math.sin(re_m)) * (Math.exp(-im) + Math.exp(im))) <= -0.002) {
		tmp = ((re_m * re_m) * -0.16666666666666666) * re_m;
	} else {
		tmp = re_m;
	}
	return re_s * tmp;
}

Python

re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
def code(re_s, re_m, im):
	tmp = 0
	if ((0.5 * math.sin(re_m)) * (math.exp(-im) + math.exp(im))) <= -0.002:
		tmp = ((re_m * re_m) * -0.16666666666666666) * re_m
	else:
		tmp = re_m
	return re_s * tmp

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.002)
		tmp = Float64(Float64(Float64(re_m * re_m) * -0.16666666666666666) * re_m);
	else
		tmp = re_m;
	end
	return Float64(re_s * tmp)
end

MATLAB

re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
function tmp_2 = code(re_s, re_m, im)
	tmp = 0.0;
	if (((0.5 * sin(re_m)) * (exp(-im) + exp(im))) <= -0.002)
		tmp = ((re_m * re_m) * -0.16666666666666666) * re_m;
	else
		tmp = re_m;
	end
	tmp_2 = re_s * tmp;
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := N[(re$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * re$95$m), $MachinePrecision], re$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -2e-3

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 29.8

         \[\leadsto \sin re \]

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 11.2

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

   8. Applied rewrites 11.2%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) \cdot re \]

       4. lift-*.f64 10.9

          \[\leadsto \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re \]

   11. Applied rewrites 10.9%

       \[\leadsto \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re \]

   if -2e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 69.4

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   5. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto re \]
   7. Step-by-step derivation

   8. Applied rewrites 34.4%

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

4. Final simplification 24.6%

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

Alternative 14: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \left(\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right)\right) \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (* re_s (* (* 0.5 (sin re_m)) (+ (exp (- im)) (exp im)))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	return re_s * ((0.5 * sin(re_m)) * (exp(-im) + exp(im)));
}

Fortran

re\_m =     private
re\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re_s, re_m, im)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im
    code = re_s * ((0.5d0 * sin(re_m)) * (exp(-im) + exp(im)))
end function

Java

re\_m = Math.abs(re);
re\_s = Math.copySign(1.0, re);
public static double code(double re_s, double re_m, double im) {
	return re_s * ((0.5 * Math.sin(re_m)) * (Math.exp(-im) + Math.exp(im)));
}

Python

re\_m = math.fabs(re)
re\_s = math.copysign(1.0, re)
def code(re_s, re_m, im):
	return re_s * ((0.5 * math.sin(re_m)) * (math.exp(-im) + math.exp(im)))

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	return Float64(re_s * Float64(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))))
end

MATLAB

re\_m = abs(re);
re\_s = sign(re) * abs(1.0);
function tmp = code(re_s, re_m, im)
	tmp = re_s * ((0.5 * sin(re_m)) * (exp(-im) + exp(im)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \]
4. Add Preprocessing

Alternative 15: 67.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\_m\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.001:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-8}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(re\_m \cdot re\_m\right) \cdot 0.004166666666666667 - 0.08333333333333333\right) \cdot re\_m, re\_m, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re_m))))
   (*
    re_s
    (if (<= t_0 -0.001)
      (* (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m) (fma im im 2.0))
      (if (<= t_0 1e-8)
        (* (* re_m 0.5) (fma (* (* im im) 0.08333333333333333) (* im im) 2.0))
        (*
         (*
          (fma
           (*
            (- (* (* re_m re_m) 0.004166666666666667) 0.08333333333333333)
            re_m)
           re_m
           0.5)
          re_m)
         (fma im im 2.0)))))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double t_0 = 0.5 * sin(re_m);
	double tmp;
	if (t_0 <= -0.001) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0);
	} else if (t_0 <= 1e-8) {
		tmp = (re_m * 0.5) * fma(((im * im) * 0.08333333333333333), (im * im), 2.0);
	} else {
		tmp = (fma(((((re_m * re_m) * 0.004166666666666667) - 0.08333333333333333) * re_m), re_m, 0.5) * re_m) * fma(im, im, 2.0);
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	t_0 = Float64(0.5 * sin(re_m))
	tmp = 0.0
	if (t_0 <= -0.001)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0));
	elseif (t_0 <= 1e-8)
		tmp = Float64(Float64(re_m * 0.5) * fma(Float64(Float64(im * im) * 0.08333333333333333), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(re_m * re_m) * 0.004166666666666667) - 0.08333333333333333) * re_m), re_m, 0.5) * re_m) * fma(im, im, 2.0));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision]}, N[(re$95$s * If[LessEqual[t$95$0, -0.001], N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-8], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * 0.004166666666666667), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * re$95$m), $MachinePrecision] * re$95$m + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -1e-3

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 69.2

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 69.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. pow2 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lift-*.f64 24.7

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 24.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if -1e-3 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 1e-8

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 100.0

         \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)} \]

   6. Taylor expanded in im around 0

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

      1. cosh-undef-rev N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \]

      2. sub0-neg N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 + \color{blue}{{im}^{2}} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2} + 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(1 + \frac{1}{12} \cdot {im}^{2}, {im}^{\color{blue}{2}}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{12} \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      8. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \frac{1}{12} + 1, {im}^{2}, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({im}^{2}, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \]

      10. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), {im}^{2}, 2\right) \]

      12. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      13. lower-*.f64 87.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 87.0%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{12} \cdot {im}^{2}, im \cdot im, 2\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \frac{1}{12}, im \cdot im, 2\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \frac{1}{12}, im \cdot im, 2\right) \]

       3. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{12}, im \cdot im, 2\right) \]

       4. lift-*.f64 87.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right) \]

   11. Applied rewrites 87.0%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right) \]

   if 1e-8 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 80.9

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 80.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      8. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      10. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      11. lift-*.f64 32.7

          \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}\right) \cdot \left(re \cdot re\right) + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}\right) \cdot \left(re \cdot re\right) + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}\right) \cdot \left(re \cdot re\right) + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}\right) \cdot \left(re \cdot re\right) + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}\right) \cdot \left(re \cdot re\right) + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\left(\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}\right) \cdot re\right) \cdot re + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}\right) \cdot re, re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}\right) \cdot re, re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) \cdot re, re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) \cdot re, re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left({re}^{2} \cdot \frac{1}{240} - \frac{1}{12}\right) \cdot re, re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left({re}^{2} \cdot \frac{1}{240} - \frac{1}{12}\right) \cdot re, re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot \frac{1}{240} - \frac{1}{12}\right) \cdot re, re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      14. lift-*.f64 32.7

          \[\leadsto \left(\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot 0.004166666666666667 - 0.08333333333333333\right) \cdot re, re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   10. Applied rewrites 32.7%

       \[\leadsto \left(\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot 0.004166666666666667 - 0.08333333333333333\right) \cdot re, re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 70.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re\_m \leq 0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right)\right) \cdot re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(re\_m \cdot re\_m\right) \cdot 0.004166666666666667, re\_m \cdot re\_m, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (*
  re_s
  (if (<= (* 0.5 (sin re_m)) 0.0005)
    (*
     (*
      (fma
       (fma
        (fma 0.002777777777777778 (* im im) 0.08333333333333333)
        (* im im)
        1.0)
       (* im im)
       2.0)
      (fma (* re_m re_m) -0.08333333333333333 0.5))
     re_m)
    (*
     (* (fma (* (* re_m re_m) 0.004166666666666667) (* re_m re_m) 0.5) re_m)
     (fma im im 2.0)))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double tmp;
	if ((0.5 * sin(re_m)) <= 0.0005) {
		tmp = (fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0) * fma((re_m * re_m), -0.08333333333333333, 0.5)) * re_m;
	} else {
		tmp = (fma(((re_m * re_m) * 0.004166666666666667), (re_m * re_m), 0.5) * re_m) * fma(im, im, 2.0);
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re_m)) <= 0.0005)
		tmp = Float64(Float64(fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0) * fma(Float64(re_m * re_m), -0.08333333333333333, 0.5)) * re_m);
	else
		tmp = Float64(Float64(fma(Float64(Float64(re_m * re_m) * 0.004166666666666667), Float64(re_m * re_m), 0.5) * re_m) * fma(im, im, 2.0));
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := N[(re$95$s * If[LessEqual[N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision], 0.0005], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * re$95$m), $MachinePrecision], N[(N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * 0.004166666666666667), $MachinePrecision] * N[(re$95$m * re$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 5.0000000000000001e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) + \frac{1}{2} \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \cdot \color{blue}{re} \]

   5. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \cosh im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]
   7. Step-by-step derivation

      1. cosh-undef-rev N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      2. sub0-neg N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      4. +-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      12. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      14. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      16. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot re \]

      17. lower-*.f64 67.3

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   8. Applied rewrites 67.3%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot re \]

   if 5.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 80.3

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 80.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      8. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      10. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      11. lift-*.f64 30.7

          \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot {re}^{2}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{240}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{240}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       3. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{240}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

       4. lift-*.f64 30.7

          \[\leadsto \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.004166666666666667, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   11. Applied rewrites 30.7%

       \[\leadsto \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.004166666666666667, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 59.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re\_m \leq 0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(re\_m \cdot re\_m\right) - 0.16666666666666666, re\_m \cdot re\_m, 1\right) \cdot re\_m\\ \end{array} \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (*
  re_s
  (if (<= (* 0.5 (sin re_m)) 0.0005)
    (* (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m) (fma im im 2.0))
    (*
     (fma
      (- (* 0.008333333333333333 (* re_m re_m)) 0.16666666666666666)
      (* re_m re_m)
      1.0)
     re_m))))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	double tmp;
	if ((0.5 * sin(re_m)) <= 0.0005) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0);
	} else {
		tmp = fma(((0.008333333333333333 * (re_m * re_m)) - 0.16666666666666666), (re_m * re_m), 1.0) * re_m;
	}
	return re_s * tmp;
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re_m)) <= 0.0005)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re_m * re_m)) - 0.16666666666666666), Float64(re_m * re_m), 1.0) * re_m);
	end
	return Float64(re_s * tmp)
end

Wolfram

re\_m = N[Abs[re], $MachinePrecision]
re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[re$95$s_, re$95$m_, im_] := N[(re$95$s * If[LessEqual[N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision], 0.0005], N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(re$95$m * re$95$m), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re$95$m * re$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 5.0000000000000001e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.0

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, \color{blue}{im}, 2\right) \]

   5. Applied rewrites 71.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. pow2 N/A

         \[\leadsto \left(\left(\frac{1}{2} + \left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      7. lift-*.f64 55.0

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 55.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 5.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 45.6

         \[\leadsto \sin re \]

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 25.0

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

   8. Applied rewrites 25.0%

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

Alternative 18: 33.8% accurate, 18.6× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re\_m \cdot re\_m, 1\right) \cdot re\_m\right) \end{array} \]

FPCore

re\_m = (fabs.f64 re)
re\_s = (copysign.f64 #s(literal 1 binary64) re)
(FPCore (re_s re_m im)
 :precision binary64
 (* re_s (* (fma -0.16666666666666666 (* re_m re_m) 1.0) re_m)))

C

re\_m = fabs(re);
re\_s = copysign(1.0, re);
double code(double re_s, double re_m, double im) {
	return re_s * (fma(-0.16666666666666666, (re_m * re_m), 1.0) * re_m);
}

Julia

re\_m = abs(re)
re\_s = copysign(1.0, re)
function code(re_s, re_m, im)
	return Float64(re_s * Float64(fma(-0.16666666666666666, Float64(re_m * re_m), 1.0) * re_m))
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. lift-sin.f64 43.8

      \[\leadsto \sin re \]

5. Applied rewrites 43.8%

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

6. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 28.8

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

8. Applied rewrites 28.8%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \color{blue}{re} \]
9. Add Preprocessing

Alternative 19: 26.2% accurate, 317.0× speedup

Math

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

FPCore

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

C

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

Fortran

re\_m =     private
re\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re_s, re_m, im)
use fmin_fmax_functions
    real(8), intent (in) :: re_s
    real(8), intent (in) :: re_m
    real(8), intent (in) :: im
    code = re_s * re_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. cosh-undef N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cosh.f64 63.8

      \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right) \]

5. Applied rewrites 63.8%

   \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)} \]

6. Taylor expanded in im around 0

   \[\leadsto re \]
7. Step-by-step derivation

8. Applied rewrites 21.1%

   \[\leadsto re \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))