math.sin on complex, real part

Percentage Accurate: 100.0% → 99.5%

Time: 3.0s

Alternatives: 17

Speedup: 1.0×

• 49.4% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := 2 \cdot \cosh im\\ t_1 := \left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right)\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(re\_m \cdot re\_m\right) \cdot -0.08333333333333333\right)\right) \cdot re\_m\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\sin re\_m \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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)) (+ (exp (- im)) (exp im)))))
   (*
    re_s
    (if (<= t_1 (- INFINITY))
      (* (* t_0 (* (* re_m re_m) -0.08333333333333333)) re_m)
      (if (<= t_1 1.0)
        (fma
         (* (sin re_m) (fma 0.041666666666666664 (* im im) 0.5))
         (* im im)
         (sin re_m))
        (* (* 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)) * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_0 * ((re_m * re_m) * -0.08333333333333333)) * re_m;
	} else if (t_1 <= 1.0) {
		tmp = fma((sin(re_m) * fma(0.041666666666666664, (im * im), 0.5)), (im * im), sin(re_m));
	} 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(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(Float64(re_m * re_m) * -0.08333333333333333)) * re_m);
	elseif (t_1 <= 1.0)
		tmp = fma(Float64(sin(re_m) * fma(0.041666666666666664, Float64(im * im), 0.5)), Float64(im * im), sin(re_m));
	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[(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$1, (-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$1, 1.0], N[(N[(N[Sin[re$95$m], $MachinePrecision] * N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[Sin[re$95$m], $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 77.6%

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

   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 14.9

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

   8. Applied rewrites 14.9%

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

   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

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \color{blue}{\sin re} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2} + \sin \color{blue}{re} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, \color{blue}{{im}^{2}}, \sin re\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {\color{blue}{im}}^{2}, \sin re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {\color{blue}{im}}^{2}, \sin re\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot \color{blue}{im}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot \color{blue}{im}, \sin re\right) \]

      13. lift-sin.f64 100.0

          \[\leadsto \mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right)} \]

   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 73.1

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

   5. Applied rewrites 73.1%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.08333333333333333\right)\right) \cdot re\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\sin re \cdot \mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

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 (fma (* im im) 0.08333333333333333 1.0) (* 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(fma((im * im), 0.08333333333333333, 1.0), (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(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(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[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 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 77.6%

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

   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 14.9

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

   8. Applied rewrites 14.9%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 73.1

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

   5. Applied rewrites 73.1%

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

4. Final simplification 70.7%

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

Alternative 3: 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 77.6%

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

   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 14.9

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

   8. Applied rewrites 14.9%

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

   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 99.9

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

   5. Applied rewrites 99.9%

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

   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 73.1

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

   5. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)} \]
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 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\_m\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re\_m \cdot re\_m, 0.004166666666666667\right) \cdot \left(re\_m \cdot re\_m\right) - 0.08333333333333333, re\_m \cdot re\_m, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \end{array} \]

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 -9.92063492063492e-5 (* re_m re_m) 0.004166666666666667)
           (* re_m re_m))
          0.08333333333333333)
         (* re_m re_m)
         0.5)
        re_m)
       (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))
      (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(-9.92063492063492e-5, (re_m * re_m), 0.004166666666666667) * (re_m * re_m)) - 0.08333333333333333), (re_m * re_m), 0.5) * re_m) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	} 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(Float64(Float64(fma(-9.92063492063492e-5, Float64(re_m * re_m), 0.004166666666666667) * Float64(re_m * re_m)) - 0.08333333333333333), Float64(re_m * re_m), 0.5) * re_m) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	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[(N[(-9.92063492063492e-5 * N[(re$95$m * re$95$m), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re$95$m * re$95$m), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re$95$m * re$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $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 im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   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 99.9

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

   5. Applied rewrites 99.9%

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

   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 73.1

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

   5. Applied rewrites 73.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right)\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re\_m \cdot re\_m, 0.004166666666666667\right) \cdot \left(re\_m \cdot re\_m\right) - 0.08333333333333333, re\_m \cdot re\_m, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \end{array} \]

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 -9.92063492063492e-5 (* re_m re_m) 0.004166666666666667)
           (* re_m re_m))
          0.08333333333333333)
         (* re_m re_m)
         0.5)
        re_m)
       (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))
      (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(-9.92063492063492e-5, (re_m * re_m), 0.004166666666666667) * (re_m * re_m)) - 0.08333333333333333), (re_m * re_m), 0.5) * re_m) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	} 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(Float64(Float64(fma(-9.92063492063492e-5, Float64(re_m * re_m), 0.004166666666666667) * Float64(re_m * re_m)) - 0.08333333333333333), Float64(re_m * re_m), 0.5) * re_m) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	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[(N[(-9.92063492063492e-5 * N[(re$95$m * re$95$m), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re$95$m * re$95$m), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re$95$m * re$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $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 im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   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 99.2

         \[\leadsto \sin re \]

   5. Applied rewrites 99.2%

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

   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 73.1

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

   5. Applied rewrites 73.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right)\\ re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re\_m \cdot re\_m, 0.004166666666666667\right) \cdot \left(re\_m \cdot re\_m\right) - 0.08333333333333333, re\_m \cdot re\_m, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, im, 2\right)\\ \end{array} \end{array} \end{array} \]

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 -9.92063492063492e-5 (* re_m re_m) 0.004166666666666667)
           (* re_m re_m))
          0.08333333333333333)
         (* re_m re_m)
         0.5)
        re_m)
       (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))
      (if (<= t_0 1.0)
        (sin re_m)
        (*
         (* re_m 0.5)
         (fma
          (*
           (fma
            (fma (* im im) 0.002777777777777778 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(-9.92063492063492e-5, (re_m * re_m), 0.004166666666666667) * (re_m * re_m)) - 0.08333333333333333), (re_m * re_m), 0.5) * re_m) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(re_m);
	} else {
		tmp = (re_m * 0.5) * fma((fma(fma((im * im), 0.002777777777777778, 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(Float64(Float64(fma(-9.92063492063492e-5, Float64(re_m * re_m), 0.004166666666666667) * Float64(re_m * re_m)) - 0.08333333333333333), Float64(re_m * re_m), 0.5) * re_m) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	elseif (t_0 <= 1.0)
		tmp = sin(re_m);
	else
		tmp = Float64(Float64(re_m * 0.5) * fma(Float64(fma(fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333), 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_] := 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[(N[(-9.92063492063492e-5 * N[(re$95$m * re$95$m), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(re$95$m * re$95$m), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re$95$m * re$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $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[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 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 im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right) \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   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 99.2

         \[\leadsto \sin re \]

   5. Applied rewrites 99.2%

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

   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 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left(\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 1\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{\color{blue}{2}}, 1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}, {im}^{2}, 1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} + \frac{1}{2}, {im}^{2}, 1\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      9. pow2 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      11. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      13. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right)\right) \]

      14. lift-*.f64 64.5

          \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \]

   8. Applied rewrites 64.5%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right)\right) \]

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 64.5

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

   11. Applied rewrites 64.5%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]
   12. 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

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

       9. lower-*.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. lower-fma.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 64.5

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

   13. Applied rewrites 64.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-9.92063492063492 \cdot 10^{-5}, re \cdot re, 0.004166666666666667\right) \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, im, 2\right)\\ \end{array} \end{array} \]

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))) 4e-5)
    (*
     (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m)
     (fma (fma (* im im) 0.08333333333333333 1.0) (* im im) 2.0))
    (*
     (* re_m 0.5)
     (fma
      (*
       (fma
        (fma (* im im) 0.002777777777777778 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))) <= 4e-5) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(fma((im * im), 0.08333333333333333, 1.0), (im * im), 2.0);
	} else {
		tmp = (re_m * 0.5) * fma((fma(fma((im * im), 0.002777777777777778, 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))) <= 4e-5)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(fma(Float64(im * im), 0.08333333333333333, 1.0), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(re_m * 0.5) * fma(Float64(fma(fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333), 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], 4e-5], N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $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))) < 4.00000000000000033e-5

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 59.4

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

   8. Applied rewrites 59.4%

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

   if 4.00000000000000033e-5 < (*.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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left(\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 1\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{\color{blue}{2}}, 1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}, {im}^{2}, 1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} + \frac{1}{2}, {im}^{2}, 1\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      9. pow2 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      11. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      13. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right)\right) \]

      14. lift-*.f64 46.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \]

   8. Applied rewrites 46.0%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right)\right) \]

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 46.0

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

   11. Applied rewrites 46.0%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]
   12. 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

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

       9. lower-*.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. lower-fma.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 46.0

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

   13. Applied rewrites 46.0%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, im, 2\right)\\ \end{array} \end{array} \]

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))) 4e-5)
    (*
     (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m)
     (fma (* (* im im) 0.08333333333333333) (* im im) 2.0))
    (*
     (* re_m 0.5)
     (fma
      (*
       (fma
        (fma (* im im) 0.002777777777777778 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))) <= 4e-5) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(((im * im) * 0.08333333333333333), (im * im), 2.0);
	} else {
		tmp = (re_m * 0.5) * fma((fma(fma((im * im), 0.002777777777777778, 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))) <= 4e-5)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(Float64(Float64(im * im) * 0.08333333333333333), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(re_m * 0.5) * fma(Float64(fma(fma(Float64(im * im), 0.002777777777777778, 0.08333333333333333), 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], 4e-5], N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $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))) < 4.00000000000000033e-5

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 59.4

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

   8. Applied rewrites 59.4%

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

   9. Taylor expanded in im around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 59.2

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

   11. Applied rewrites 59.2%

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

   if 4.00000000000000033e-5 < (*.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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left(\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 1\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{\color{blue}{2}}, 1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}, {im}^{2}, 1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} + \frac{1}{2}, {im}^{2}, 1\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      9. pow2 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      11. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      13. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right)\right) \]

      14. lift-*.f64 46.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \]

   8. Applied rewrites 46.0%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right)\right) \]

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 46.0

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

   11. Applied rewrites 46.0%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]
   12. 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

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

       9. lower-*.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. lower-fma.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 46.0

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

   13. Applied rewrites 46.0%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.002777777777777778, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

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))) 4e-5)
    (*
     (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m)
     (fma (* (* im im) 0.08333333333333333) (* im im) 2.0))
    (*
     (* re_m 0.5)
     (fma
      (fma (* (* im im) 0.002777777777777778) (* 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))) <= 4e-5) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(((im * im) * 0.08333333333333333), (im * im), 2.0);
	} else {
		tmp = (re_m * 0.5) * fma(fma(((im * im) * 0.002777777777777778), (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))) <= 4e-5)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(Float64(Float64(im * im) * 0.08333333333333333), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(re_m * 0.5) * fma(fma(Float64(Float64(im * im) * 0.002777777777777778), Float64(im * im), 1.0), 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], 4e-5], N[(N[(N[(N[(re$95$m * re$95$m), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re$95$m), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re$95$m * 0.5), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $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))) < 4.00000000000000033e-5

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 59.4

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

   8. Applied rewrites 59.4%

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

   9. Taylor expanded in im around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 59.2

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

   11. Applied rewrites 59.2%

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

   if 4.00000000000000033e-5 < (*.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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left(\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 1\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{\color{blue}{2}}, 1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}, {im}^{2}, 1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} + \frac{1}{2}, {im}^{2}, 1\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      9. pow2 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      11. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      13. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right)\right) \]

      14. lift-*.f64 46.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \]

   8. Applied rewrites 46.0%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right)\right) \]

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 46.0

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

   11. Applied rewrites 46.0%

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

   12. Taylor expanded in im around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 46.0

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

   14. Applied rewrites 46.0%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re\_m \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

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))) 4e-5)
    (* (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m) (fma im im 2.0))
    (*
     (* re_m 0.5)
     (fma
      (fma (* (* im im) 0.002777777777777778) (* 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))) <= 4e-5) {
		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) * 0.002777777777777778), (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))) <= 4e-5)
		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(fma(Float64(Float64(im * im) * 0.002777777777777778), Float64(im * im), 1.0), 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], 4e-5], 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] * 0.002777777777777778), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $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))) < 4.00000000000000033e-5

   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 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 54.6

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

   8. Applied rewrites 54.6%

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

   if 4.00000000000000033e-5 < (*.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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \left(\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 1\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{\color{blue}{2}}, 1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}, {im}^{2}, 1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2} + \frac{1}{2}, {im}^{2}, 1\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      9. pow2 N/A

         \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      11. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), {im}^{2}, 1\right)\right) \]

      13. pow2 N/A

          \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right)\right) \]

      14. lift-*.f64 46.0

          \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\right) \]

   8. Applied rewrites 46.0%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \left(2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right)\right) \]

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 46.0

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

   11. Applied rewrites 46.0%

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

   12. Taylor expanded in im around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 46.0

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

   14. Applied rewrites 46.0%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(re\_m \cdot re\_m, -0.08333333333333333, 0.5\right) \cdot re\_m\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re\_m\\ \end{array} \end{array} \]

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)) (+ (exp (- im)) (exp im))) 4e-5)
    (* (* (fma (* re_m re_m) -0.08333333333333333 0.5) re_m) (fma im im 2.0))
    (* (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 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)) * (exp(-im) + exp(im))) <= 4e-5) {
		tmp = (fma((re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0);
	} else {
		tmp = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 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(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))) <= 4e-5)
		tmp = Float64(Float64(fma(Float64(re_m * re_m), -0.08333333333333333, 0.5) * re_m) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 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[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], 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[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re$95$m), $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))) < 4.00000000000000033e-5

   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 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 54.6

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

   8. Applied rewrites 54.6%

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

   if 4.00000000000000033e-5 < (*.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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re, {im}^{\color{blue}{2}}, re\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot re\right) \cdot \frac{1}{24} + \frac{1}{2} \cdot re, {im}^{2}, re\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, re \cdot \frac{1}{2}\right), {im}^{2}, re\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, re \cdot \frac{1}{2}\right), {im}^{2}, re\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, re \cdot \frac{1}{2}\right), im \cdot im, re\right) \]

      12. lift-*.f64 40.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.041666666666666664, re \cdot 0.5\right), im \cdot im, re\right) \]

   8. Applied rewrites 40.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.041666666666666664, re \cdot 0.5\right), \color{blue}{im \cdot im}, re\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

       2. lower-*.f64 N/A

          \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

       3. +-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot re \]

       4. pow2 N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot re \]

       5. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot re \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, im \cdot im, 1\right) \cdot re \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}, im \cdot im, 1\right) \cdot re \]

       8. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) + \frac{1}{2}, im \cdot im, 1\right) \cdot re \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{24} + \frac{1}{2}, im \cdot im, 1\right) \cdot re \]

       10. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), im \cdot im, 1\right) \cdot re \]

       11. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), im \cdot im, 1\right) \cdot re \]

       12. lift-*.f64 44.0

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re \]

   11. Applied rewrites 44.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re\_m \cdot re\_m, 1\right) \cdot re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re\_m\\ \end{array} \end{array} \]

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)) (+ (exp (- im)) (exp im))) 4e-5)
    (* (fma -0.16666666666666666 (* re_m re_m) 1.0) re_m)
    (* (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 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)) * (exp(-im) + exp(im))) <= 4e-5) {
		tmp = fma(-0.16666666666666666, (re_m * re_m), 1.0) * re_m;
	} else {
		tmp = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 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(Float64(0.5 * sin(re_m)) * Float64(exp(Float64(-im)) + exp(im))) <= 4e-5)
		tmp = Float64(fma(-0.16666666666666666, Float64(re_m * re_m), 1.0) * re_m);
	else
		tmp = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 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[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], N[(N[(-0.16666666666666666 * N[(re$95$m * re$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * re$95$m), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re$95$m), $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))) < 4.00000000000000033e-5

   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 58.6

         \[\leadsto \sin re \]

   5. Applied rewrites 58.6%

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

   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 42.1

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

   8. Applied rewrites 42.1%

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

   if 4.00000000000000033e-5 < (*.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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re, {im}^{\color{blue}{2}}, re\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot re\right) \cdot \frac{1}{24} + \frac{1}{2} \cdot re, {im}^{2}, re\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, re \cdot \frac{1}{2}\right), {im}^{2}, re\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, re \cdot \frac{1}{2}\right), {im}^{2}, re\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, re \cdot \frac{1}{2}\right), im \cdot im, re\right) \]

      12. lift-*.f64 40.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.041666666666666664, re \cdot 0.5\right), im \cdot im, re\right) \]

   8. Applied rewrites 40.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.041666666666666664, re \cdot 0.5\right), \color{blue}{im \cdot im}, re\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto re \cdot \left(1 + \color{blue}{{im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

       2. lower-*.f64 N/A

          \[\leadsto \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) \cdot re \]

       3. +-commutative N/A

          \[\leadsto \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot re \]

       4. pow2 N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right) \cdot re \]

       5. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot re \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, im \cdot im, 1\right) \cdot re \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}, im \cdot im, 1\right) \cdot re \]

       8. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) + \frac{1}{2}, im \cdot im, 1\right) \cdot re \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{24} + \frac{1}{2}, im \cdot im, 1\right) \cdot re \]

       10. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), im \cdot im, 1\right) \cdot re \]

       11. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{24}, \frac{1}{2}\right), im \cdot im, 1\right) \cdot re \]

       12. lift-*.f64 44.0

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re \]

   11. Applied rewrites 44.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re\_m \cdot re\_m, 1\right) \cdot re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\_m\right) \cdot 0.041666666666666664, im \cdot im, re\_m\right)\\ \end{array} \end{array} \]

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))) 4e-5)
    (* (fma -0.16666666666666666 (* re_m re_m) 1.0) re_m)
    (fma (* (* (* im im) re_m) 0.041666666666666664) (* im im) 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))) <= 4e-5) {
		tmp = fma(-0.16666666666666666, (re_m * re_m), 1.0) * re_m;
	} else {
		tmp = fma((((im * im) * re_m) * 0.041666666666666664), (im * im), 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))) <= 4e-5)
		tmp = Float64(fma(-0.16666666666666666, Float64(re_m * re_m), 1.0) * re_m);
	else
		tmp = fma(Float64(Float64(Float64(im * im) * re_m) * 0.041666666666666664), Float64(im * im), 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[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], N[(N[(-0.16666666666666666 * N[(re$95$m * re$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * re$95$m), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * re$95$m), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + re$95$m), $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))) < 4.00000000000000033e-5

   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 58.6

         \[\leadsto \sin re \]

   5. Applied rewrites 58.6%

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

   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 42.1

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

   8. Applied rewrites 42.1%

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

   if 4.00000000000000033e-5 < (*.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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto re + \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) + re \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re\right) \cdot {im}^{2} + re \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right) + \frac{1}{2} \cdot re, {im}^{\color{blue}{2}}, re\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot re\right) \cdot \frac{1}{24} + \frac{1}{2} \cdot re, {im}^{2}, re\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, \frac{1}{2} \cdot re\right), {im}^{2}, re\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, re \cdot \frac{1}{2}\right), {im}^{2}, re\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, re \cdot \frac{1}{2}\right), {im}^{2}, re\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, \frac{1}{24}, re \cdot \frac{1}{2}\right), im \cdot im, re\right) \]

      12. lift-*.f64 40.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.041666666666666664, re \cdot 0.5\right), im \cdot im, re\right) \]

   8. Applied rewrites 40.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.041666666666666664, re \cdot 0.5\right), \color{blue}{im \cdot im}, re\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot re\right), im \cdot im, re\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot re\right) \cdot \frac{1}{24}, im \cdot im, re\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left({im}^{2} \cdot re\right) \cdot \frac{1}{24}, im \cdot im, re\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot \frac{1}{24}, im \cdot im, re\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot \frac{1}{24}, im \cdot im, re\right) \]

       5. lift-*.f64 40.1

          \[\leadsto \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.041666666666666664, im \cdot im, re\right) \]

   11. Applied rewrites 40.1%

       \[\leadsto \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.041666666666666664, im \cdot im, re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.041666666666666664, im \cdot im, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} re\_m = \left|re\right| \\ re\_s = \mathsf{copysign}\left(1, re\right) \\ re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\_m\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re\_m \cdot re\_m, 1\right) \cdot re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot re\_m, 0.5, re\_m\right)\\ \end{array} \end{array} \]

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))) 4e-5)
    (* (fma -0.16666666666666666 (* re_m re_m) 1.0) re_m)
    (fma (* (* im im) re_m) 0.5 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))) <= 4e-5) {
		tmp = fma(-0.16666666666666666, (re_m * re_m), 1.0) * re_m;
	} else {
		tmp = fma(((im * im) * re_m), 0.5, 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))) <= 4e-5)
		tmp = Float64(fma(-0.16666666666666666, Float64(re_m * re_m), 1.0) * re_m);
	else
		tmp = fma(Float64(Float64(im * im) * re_m), 0.5, 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[(N[(0.5 * N[Sin[re$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], N[(N[(-0.16666666666666666 * N[(re$95$m * re$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * re$95$m), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * re$95$m), $MachinePrecision] * 0.5 + re$95$m), $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))) < 4.00000000000000033e-5

   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 58.6

         \[\leadsto \sin re \]

   5. Applied rewrites 58.6%

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

   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 42.1

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

   8. Applied rewrites 42.1%

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

   if 4.00000000000000033e-5 < (*.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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 35.1

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

   8. Applied rewrites 35.1%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, 0.5, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 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 16: 34.3% 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 48.7

      \[\leadsto \sin re \]

5. Applied rewrites 48.7%

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

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 31.3

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

8. Applied rewrites 31.3%

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

Alternative 17: 26.3% 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 65.5

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

5. Applied rewrites 65.5%

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

6. Taylor expanded in im around 0

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

8. Applied rewrites 25.3%

   \[\leadsto re \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025083 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))