math.sin on complex, imaginary part

Percentage Accurate: 54.5% → 99.5%

Time: 9.4s

Alternatives: 18

Speedup: 1.5×

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

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im - \sinh im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (* 0.5 (cos re))
  (-
   (*
    (fma
     (fma
      (fma -0.0001984126984126984 (* im im) -0.008333333333333333)
      (* im im)
      -0.16666666666666666)
     (* im im)
     -1.0)
    im)
   (sinh im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * ((fma(fma(fma(-0.0001984126984126984, (im * im), -0.008333333333333333), (im * im), -0.16666666666666666), (im * im), -1.0) * im) - sinh(im));
}

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(im * im), -0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), -1.0) * im) - sinh(im)))
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision] * im), $MachinePrecision] - N[Sinh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

   2. lift-exp.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{e^{im}}\right) \]

   3. sinh-+-cosh-rev N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{\left(\cosh im + \sinh im\right)}\right) \]

   4. associate--r+ N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(e^{0 - im} - \cosh im\right)} - \sinh im\right) \]

   7. lift--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{0 - im}} - \cosh im\right) - \sinh im\right) \]

   8. sub0-neg N/A

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

   9. lower-neg.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{-im}} - \cosh im\right) - \sinh im\right) \]

   10. lower-cosh.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{-im} - \color{blue}{\cosh im}\right) - \sinh im\right) \]

   11. lower-sinh.f64 38.5

       \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(e^{-im} - \cosh im\right) - \color{blue}{\sinh im}\right) \]

4. Applied rewrites 38.5%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{-im} - \cosh im\right) - \sinh im\right)} \]

5. Taylor expanded in im around 0

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

7. Applied rewrites 99.8%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im} - \sinh im\right) \]
8. Add Preprocessing

Alternative 2: 87.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im - \sinh im\\ t_1 := 0.5 \cdot \cos re\\ t_2 := t\_1 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (-
          (*
           (fma
            (fma
             (fma -0.0001984126984126984 (* im im) -0.008333333333333333)
             (* im im)
             -0.16666666666666666)
            (* im im)
            -1.0)
           im)
          (sinh im)))
        (t_1 (* 0.5 (cos re)))
        (t_2 (* t_1 (- (exp (- im)) (exp im)))))
   (if (<= t_2 (- INFINITY))
     (* 0.5 t_0)
     (if (<= t_2 0.4)
       (*
        t_1
        (*
         (fma
          (fma
           (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
           im
           -0.3333333333333333)
          (* im im)
          -2.0)
         im))
       (* (fma (* re re) -0.25 0.5) t_0)))))

C

double code(double re, double im) {
	double t_0 = (fma(fma(fma(-0.0001984126984126984, (im * im), -0.008333333333333333), (im * im), -0.16666666666666666), (im * im), -1.0) * im) - sinh(im);
	double t_1 = 0.5 * cos(re);
	double t_2 = t_1 * (exp(-im) - exp(im));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = 0.5 * t_0;
	} else if (t_2 <= 0.4) {
		tmp = t_1 * (fma(fma((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im), im, -0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = fma((re * re), -0.25, 0.5) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(im * im), -0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), -1.0) * im) - sinh(im))
	t_1 = Float64(0.5 * cos(re))
	t_2 = Float64(t_1 * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(0.5 * t_0);
	elseif (t_2 <= 0.4)
		tmp = Float64(t_1 * Float64(fma(fma(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im), im, -0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision] * im), $MachinePrecision] - N[Sinh[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(t$95$1 * N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

      2. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{e^{im}}\right) \]

      3. sinh-+-cosh-rev N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{\left(\cosh im + \sinh im\right)}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(e^{0 - im} - \cosh im\right)} - \sinh im\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{0 - im}} - \cosh im\right) - \sinh im\right) \]

      8. sub0-neg N/A

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

      9. lower-neg.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{-im}} - \cosh im\right) - \sinh im\right) \]

      10. lower-cosh.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{-im} - \color{blue}{\cosh im}\right) - \sinh im\right) \]

      11. lower-sinh.f64 73.4

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(e^{-im} - \cosh im\right) - \color{blue}{\sinh im}\right) \]

   4. Applied rewrites 73.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{-im} - \cosh im\right) - \sinh im\right)} \]

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, im \cdot im, \frac{-1}{120}\right), im \cdot im, \frac{-1}{6}\right), im \cdot im, -1\right) \cdot im - \sinh im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.4%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.40000000000000002

   1. Initial program 7.6%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   if 0.40000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

      2. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{e^{im}}\right) \]

      3. sinh-+-cosh-rev N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{\left(\cosh im + \sinh im\right)}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(e^{0 - im} - \cosh im\right)} - \sinh im\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{0 - im}} - \cosh im\right) - \sinh im\right) \]

      8. sub0-neg N/A

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

      9. lower-neg.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{-im}} - \cosh im\right) - \sinh im\right) \]

      10. lower-cosh.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{-im} - \color{blue}{\cosh im}\right) - \sinh im\right) \]

      11. lower-sinh.f64 38.8

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(e^{-im} - \cosh im\right) - \color{blue}{\sinh im}\right) \]

   4. Applied rewrites 38.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{-im} - \cosh im\right) - \sinh im\right)} \]

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, im \cdot im, \frac{-1}{120}\right), im \cdot im, \frac{-1}{6}\right), im \cdot im, -1\right) \cdot im - \sinh im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im - \sinh im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im - \sinh im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im - \sinh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\\ t_1 := 0.5 \cdot \cos re\\ t_2 := t\_1 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im - \sinh im\right)\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_1 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma
            (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
            im
            -0.3333333333333333)
           (* im im)
           -2.0)
          im))
        (t_1 (* 0.5 (cos re)))
        (t_2 (* t_1 (- (exp (- im)) (exp im)))))
   (if (<= t_2 (- INFINITY))
     (*
      0.5
      (-
       (*
        (fma
         (fma
          (fma -0.0001984126984126984 (* im im) -0.008333333333333333)
          (* im im)
          -0.16666666666666666)
         (* im im)
         -1.0)
        im)
       (sinh im)))
     (if (<= t_2 0.4)
       (* t_1 t_0)
       (*
        (fma
         (-
          (*
           (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
           re)
          0.25)
         (* re re)
         0.5)
        t_0)))))

C

double code(double re, double im) {
	double t_0 = fma(fma((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im), im, -0.3333333333333333), (im * im), -2.0) * im;
	double t_1 = 0.5 * cos(re);
	double t_2 = t_1 * (exp(-im) - exp(im));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = 0.5 * ((fma(fma(fma(-0.0001984126984126984, (im * im), -0.008333333333333333), (im * im), -0.16666666666666666), (im * im), -1.0) * im) - sinh(im));
	} else if (t_2 <= 0.4) {
		tmp = t_1 * t_0;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(fma(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im), im, -0.3333333333333333), Float64(im * im), -2.0) * im)
	t_1 = Float64(0.5 * cos(re))
	t_2 = Float64(t_1 * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(im * im), -0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), -1.0) * im) - sinh(im)));
	elseif (t_2 <= 0.4)
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(0.5 * N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision] * im), $MachinePrecision] - N[Sinh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(t$95$1 * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

      2. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{e^{im}}\right) \]

      3. sinh-+-cosh-rev N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{\left(\cosh im + \sinh im\right)}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(e^{0 - im} - \cosh im\right)} - \sinh im\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{0 - im}} - \cosh im\right) - \sinh im\right) \]

      8. sub0-neg N/A

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

      9. lower-neg.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{-im}} - \cosh im\right) - \sinh im\right) \]

      10. lower-cosh.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{-im} - \color{blue}{\cosh im}\right) - \sinh im\right) \]

      11. lower-sinh.f64 73.4

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(e^{-im} - \cosh im\right) - \color{blue}{\sinh im}\right) \]

   4. Applied rewrites 73.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{-im} - \cosh im\right) - \sinh im\right)} \]

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, im \cdot im, \frac{-1}{120}\right), im \cdot im, \frac{-1}{6}\right), im \cdot im, -1\right) \cdot im - \sinh im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.4%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.40000000000000002

   1. Initial program 7.6%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   if 0.40000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 70.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im - \sinh im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im - \sinh im\right)\\ \mathbf{elif}\;t\_1 \leq 0.4:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (- (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      0.5
      (-
       (*
        (fma
         (fma
          (fma -0.0001984126984126984 (* im im) -0.008333333333333333)
          (* im im)
          -0.16666666666666666)
         (* im im)
         -1.0)
        im)
       (sinh im)))
     (if (<= t_1 0.4)
       (*
        t_0
        (*
         (fma
          (fma -0.016666666666666666 (* im im) -0.3333333333333333)
          (* im im)
          -2.0)
         im))
       (*
        (fma
         (-
          (*
           (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
           re)
          0.25)
         (* re re)
         0.5)
        (*
         (fma
          (fma
           (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
           im
           -0.3333333333333333)
          (* im im)
          -2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) - exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.5 * ((fma(fma(fma(-0.0001984126984126984, (im * im), -0.008333333333333333), (im * im), -0.16666666666666666), (im * im), -1.0) * im) - sinh(im));
	} else if (t_1 <= 0.4) {
		tmp = t_0 * (fma(fma(-0.016666666666666666, (im * im), -0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * (fma(fma((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im), im, -0.3333333333333333), (im * im), -2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(im * im), -0.008333333333333333), Float64(im * im), -0.16666666666666666), Float64(im * im), -1.0) * im) - sinh(im)));
	elseif (t_1 <= 0.4)
		tmp = Float64(t_0 * Float64(fma(fma(-0.016666666666666666, Float64(im * im), -0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(fma(fma(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im), im, -0.3333333333333333), Float64(im * im), -2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(0.5 * N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision] * im), $MachinePrecision] - N[Sinh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.4], N[(t$95$0 * N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

      2. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{e^{im}}\right) \]

      3. sinh-+-cosh-rev N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{\left(\cosh im + \sinh im\right)}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(e^{0 - im} - \cosh im\right)} - \sinh im\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{0 - im}} - \cosh im\right) - \sinh im\right) \]

      8. sub0-neg N/A

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

      9. lower-neg.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{-im}} - \cosh im\right) - \sinh im\right) \]

      10. lower-cosh.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{-im} - \color{blue}{\cosh im}\right) - \sinh im\right) \]

      11. lower-sinh.f64 73.4

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(e^{-im} - \cosh im\right) - \color{blue}{\sinh im}\right) \]

   4. Applied rewrites 73.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{-im} - \cosh im\right) - \sinh im\right)} \]

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, im \cdot im, \frac{-1}{120}\right), im \cdot im, \frac{-1}{6}\right), im \cdot im, -1\right) \cdot im - \sinh im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.4%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.40000000000000002

   1. Initial program 7.6%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if 0.40000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 70.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im - \sinh im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} - e^{im}\right)\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 0.4:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re)))
        (t_1 (* t_0 (- (exp (- im)) (exp im))))
        (t_2
         (*
          (fma
           (fma
            (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
            im
            -0.3333333333333333)
           (* im im)
           -2.0)
          im)))
   (if (<= t_1 (- INFINITY))
     (* (fma (- (* 0.020833333333333332 (* re re)) 0.25) (* re re) 0.5) t_2)
     (if (<= t_1 0.4)
       (*
        t_0
        (*
         (fma
          (fma -0.016666666666666666 (* im im) -0.3333333333333333)
          (* im im)
          -2.0)
         im))
       (*
        (fma
         (-
          (*
           (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
           re)
          0.25)
         (* re re)
         0.5)
        t_2)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) - exp(im));
	double t_2 = fma(fma((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im), im, -0.3333333333333333), (im * im), -2.0) * im;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((0.020833333333333332 * (re * re)) - 0.25), (re * re), 0.5) * t_2;
	} else if (t_1 <= 0.4) {
		tmp = t_0 * (fma(fma(-0.016666666666666666, (im * im), -0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) - exp(im)))
	t_2 = Float64(fma(fma(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im), im, -0.3333333333333333), Float64(im * im), -2.0) * im)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(0.020833333333333332 * Float64(re * re)) - 0.25), Float64(re * re), 0.5) * t_2);
	elseif (t_1 <= 0.4)
		tmp = Float64(t_0 * Float64(fma(fma(-0.016666666666666666, Float64(im * im), -0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * t_2);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 0.4], N[(t$95$0 * N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 83.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 64.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.40000000000000002

   1. Initial program 7.6%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if 0.40000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 70.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0.4:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))))
        (t_1
         (*
          (fma
           (fma
            (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
            im
            -0.3333333333333333)
           (* im im)
           -2.0)
          im)))
   (if (<= t_0 (- INFINITY))
     (* (fma (- (* 0.020833333333333332 (* re re)) 0.25) (* re re) 0.5) t_1)
     (if (<= t_0 0.4)
       (* (* (cos re) (fma (* im im) -0.16666666666666666 -1.0)) im)
       (*
        (fma
         (-
          (*
           (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
           re)
          0.25)
         (* re re)
         0.5)
        t_1)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) - exp(im));
	double t_1 = fma(fma((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im), im, -0.3333333333333333), (im * im), -2.0) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((0.020833333333333332 * (re * re)) - 0.25), (re * re), 0.5) * t_1;
	} else if (t_0 <= 0.4) {
		tmp = (cos(re) * fma((im * im), -0.16666666666666666, -1.0)) * im;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(fma(fma(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im), im, -0.3333333333333333), Float64(im * im), -2.0) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(0.020833333333333332 * Float64(re * re)) - 0.25), Float64(re * re), 0.5) * t_1);
	elseif (t_0 <= 0.4)
		tmp = Float64(Float64(cos(re) * fma(Float64(im * im), -0.16666666666666666, -1.0)) * im);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * t_1);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.4], N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 83.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 64.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.40000000000000002

   1. Initial program 7.6%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 99.3%

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

   if 0.40000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 70.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.4:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0.4:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))))
        (t_1
         (*
          (fma
           (fma
            (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
            im
            -0.3333333333333333)
           (* im im)
           -2.0)
          im)))
   (if (<= t_0 (- INFINITY))
     (* (fma (- (* 0.020833333333333332 (* re re)) 0.25) (* re re) 0.5) t_1)
     (if (<= t_0 0.4)
       (* (- (cos re)) im)
       (*
        (fma
         (-
          (*
           (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
           re)
          0.25)
         (* re re)
         0.5)
        t_1)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) - exp(im));
	double t_1 = fma(fma((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im), im, -0.3333333333333333), (im * im), -2.0) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((0.020833333333333332 * (re * re)) - 0.25), (re * re), 0.5) * t_1;
	} else if (t_0 <= 0.4) {
		tmp = -cos(re) * im;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(fma(fma(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im), im, -0.3333333333333333), Float64(im * im), -2.0) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(0.020833333333333332 * Float64(re * re)) - 0.25), Float64(re * re), 0.5) * t_1);
	elseif (t_0 <= 0.4)
		tmp = Float64(Float64(-cos(re)) * im);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * t_1);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.4], N[((-N[Cos[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 83.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 64.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.40000000000000002

   1. Initial program 7.6%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.1%

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

   if 0.40000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 70.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.4:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\\ \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma
            (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
            im
            -0.3333333333333333)
           (* im im)
           -2.0)
          im)))
   (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
     (* 0.5 t_0)
     (*
      (fma
       (-
        (*
         (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
         re)
        0.25)
       (* re re)
       0.5)
      t_0))))

C

double code(double re, double im) {
	double t_0 = fma(fma((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im), im, -0.3333333333333333), (im * im), -2.0) * im;
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = 0.5 * t_0;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(fma(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im), im, -0.3333333333333333), Float64(im * im), -2.0) * im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.8%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 54.6%

       \[\leadsto \color{blue}{0.5} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 70.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (*
    0.5
    (*
     (fma
      (fma
       (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
       im
       -0.3333333333333333)
      (* im im)
      -2.0)
     im))
   (*
    (fma
     (-
      (* (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re) re)
      0.25)
     (* re re)
     0.5)
    (*
     (fma
      (- (* -0.016666666666666666 (* im im)) 0.3333333333333333)
      (* im im)
      -2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = 0.5 * (fma(fma((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im), im, -0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * (fma(((-0.016666666666666666 * (im * im)) - 0.3333333333333333), (im * im), -2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(0.5 * Float64(fma(fma(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im), im, -0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(fma(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333), Float64(im * im), -2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.8%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 54.6%

       \[\leadsto \color{blue}{0.5} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 83.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 68.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (*
    0.5
    (*
     (fma
      (fma
       (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
       im
       -0.3333333333333333)
      (* im im)
      -2.0)
     im))
   (*
    (*
     (fma
      (fma
       (fma -0.001388888888888889 (* re re) 0.041666666666666664)
       (* re re)
       -0.5)
      (* re re)
      1.0)
     (fma -0.16666666666666666 (* im im) -1.0))
    im)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = 0.5 * (fma(fma((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im), im, -0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = (fma(fma(fma(-0.001388888888888889, (re * re), 0.041666666666666664), (re * re), -0.5), (re * re), 1.0) * fma(-0.16666666666666666, (im * im), -1.0)) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(0.5 * Float64(fma(fma(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im), im, -0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), -0.5), Float64(re * re), 1.0) * fma(-0.16666666666666666, Float64(im * im), -1.0)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.8%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 54.6%

       \[\leadsto \color{blue}{0.5} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

      2. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{e^{im}}\right) \]

      3. sinh-+-cosh-rev N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{\left(\cosh im + \sinh im\right)}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(e^{0 - im} - \cosh im\right)} - \sinh im\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{0 - im}} - \cosh im\right) - \sinh im\right) \]

      8. sub0-neg N/A

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

      9. lower-neg.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{-im}} - \cosh im\right) - \sinh im\right) \]

      10. lower-cosh.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{-im} - \color{blue}{\cosh im}\right) - \sinh im\right) \]

      11. lower-sinh.f64 39.0

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(e^{-im} - \cosh im\right) - \color{blue}{\sinh im}\right) \]

   4. Applied rewrites 39.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{-im} - \cosh im\right) - \sinh im\right)} \]

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 69.3%

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

   11. Taylor expanded in re around 0

       \[\leadsto \left(\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, im \cdot im, -1\right)\right) \cdot im \]
   12. Step-by-step derivation

   13. Applied rewrites 62.5%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (*
    0.5
    (*
     (fma
      (fma
       (* (fma -0.0003968253968253968 (* im im) -0.016666666666666666) im)
       im
       -0.3333333333333333)
      (* im im)
      -2.0)
     im))
   (*
    (fma
     (fma
      (fma -0.001388888888888889 (* re re) 0.041666666666666664)
      (* re re)
      -0.5)
     (* re re)
     1.0)
    (- im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = 0.5 * (fma(fma((fma(-0.0003968253968253968, (im * im), -0.016666666666666666) * im), im, -0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = fma(fma(fma(-0.001388888888888889, (re * re), 0.041666666666666664), (re * re), -0.5), (re * re), 1.0) * -im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(0.5 * Float64(fma(fma(Float64(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666) * im), im, -0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(fma(fma(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), -0.5), Float64(re * re), 1.0) * Float64(-im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.8%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im, im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 54.6%

       \[\leadsto \color{blue}{0.5} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.2%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 37.7%

      \[\leadsto \left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right)\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(-im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (*
    0.5
    (*
     (fma
      (- (* -0.016666666666666666 (* im im)) 0.3333333333333333)
      (* im im)
      -2.0)
     im))
   (*
    (fma
     (fma
      (fma -0.001388888888888889 (* re re) 0.041666666666666664)
      (* re re)
      -0.5)
     (* re re)
     1.0)
    (- im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = 0.5 * (fma(((-0.016666666666666666 * (im * im)) - 0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = fma(fma(fma(-0.001388888888888889, (re * re), 0.041666666666666664), (re * re), -0.5), (re * re), 1.0) * -im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(0.5 * Float64(fma(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(fma(fma(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), -0.5), Float64(re * re), 1.0) * Float64(-im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.8%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 53.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.2%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 37.7%

      \[\leadsto \left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right)\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(-im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (*
    0.5
    (*
     (fma
      (- (* -0.016666666666666666 (* im im)) 0.3333333333333333)
      (* im im)
      -2.0)
     im))
   (* (fma (* re re) -0.25 0.5) (* (* -0.3333333333333333 (* im im)) im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = 0.5 * (fma(((-0.016666666666666666 * (im * im)) - 0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = fma((re * re), -0.25, 0.5) * ((-0.3333333333333333 * (im * im)) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(0.5 * Float64(fma(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(Float64(-0.3333333333333333 * Float64(im * im)) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.8%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 53.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 83.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.3%

       \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \color{blue}{\left(\left(\left(-0.3333333333333333 \cdot im\right) \cdot im - 2\right) \cdot im\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2}\right) \cdot im\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 57.0%

       \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (* (fma (* im im) -0.16666666666666666 -1.0) im)
   (* (fma (* re re) -0.25 0.5) (* (* -0.3333333333333333 (* im im)) im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = fma((im * im), -0.16666666666666666, -1.0) * im;
	} else {
		tmp = fma((re * re), -0.25, 0.5) * ((-0.3333333333333333 * (im * im)) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, -1.0) * im);
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(Float64(-0.3333333333333333 * Float64(im * im)) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

      2. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{e^{im}}\right) \]

      3. sinh-+-cosh-rev N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{\left(\cosh im + \sinh im\right)}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(e^{0 - im} - \cosh im\right)} - \sinh im\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{0 - im}} - \cosh im\right) - \sinh im\right) \]

      8. sub0-neg N/A

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

      9. lower-neg.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{-im}} - \cosh im\right) - \sinh im\right) \]

      10. lower-cosh.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{-im} - \color{blue}{\cosh im}\right) - \sinh im\right) \]

      11. lower-sinh.f64 38.4

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(e^{-im} - \cosh im\right) - \color{blue}{\sinh im}\right) \]

   4. Applied rewrites 38.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{-im} - \cosh im\right) - \sinh im\right)} \]

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 87.5%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 50.6%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 83.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.3%

       \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \color{blue}{\left(\left(\left(-0.3333333333333333 \cdot im\right) \cdot im - 2\right) \cdot im\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2}\right) \cdot im\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 57.0%

       \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (* (fma (* im im) -0.16666666666666666 -1.0) im)
   (* (* (* re re) 0.5) im)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = fma((im * im), -0.16666666666666666, -1.0) * im;
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

      2. lift-exp.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{e^{im}}\right) \]

      3. sinh-+-cosh-rev N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{\left(\cosh im + \sinh im\right)}\right) \]

      4. associate--r+ N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(e^{0 - im} - \cosh im\right)} - \sinh im\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{0 - im}} - \cosh im\right) - \sinh im\right) \]

      8. sub0-neg N/A

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

      9. lower-neg.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{-im}} - \cosh im\right) - \sinh im\right) \]

      10. lower-cosh.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{-im} - \color{blue}{\cosh im}\right) - \sinh im\right) \]

      11. lower-sinh.f64 38.4

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(e^{-im} - \cosh im\right) - \color{blue}{\sinh im}\right) \]

   4. Applied rewrites 38.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{-im} - \cosh im\right) - \sinh im\right)} \]

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 87.5%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 50.6%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.2%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.4%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 26.4%

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

4. Final simplification 44.2%

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

Alternative 16: 33.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (- im)
   (* (* (* re re) 0.5) im)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = -im;
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im) - exp(im))) <= 0.0d0) then
        tmp = -im
    else
        tmp = ((re * re) * 0.5d0) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im) - Math.exp(im))) <= 0.0) {
		tmp = -im;
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im) - math.exp(im))) <= 0.0:
		tmp = -im
	else:
		tmp = ((re * re) * 0.5) * im
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0)
		tmp = -im;
	else
		tmp = ((re * re) * 0.5) * im;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.8%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto -1 \cdot \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.5%

      \[\leadsto -im \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.2%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.4%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 26.4%

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

4. Final simplification 32.4%

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

Alternative 17: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(\left(-im\right) - \sinh im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 (cos re)) (- (- im) (sinh im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (-im - sinh(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 * cos(re)) * (-im - sinh(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (-im - Math.sinh(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (-im - math.sinh(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(Float64(-im) - sinh(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (-im - sinh(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[((-im) - N[Sinh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

   2. lift-exp.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{e^{im}}\right) \]

   3. sinh-+-cosh-rev N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{\left(\cosh im + \sinh im\right)}\right) \]

   4. associate--r+ N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{0 - im} - \cosh im\right) - \sinh im\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(e^{0 - im} - \cosh im\right)} - \sinh im\right) \]

   7. lift--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{0 - im}} - \cosh im\right) - \sinh im\right) \]

   8. sub0-neg N/A

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

   9. lower-neg.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{\color{blue}{-im}} - \cosh im\right) - \sinh im\right) \]

   10. lower-cosh.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(e^{-im} - \color{blue}{\cosh im}\right) - \sinh im\right) \]

   11. lower-sinh.f64 38.5

       \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(e^{-im} - \cosh im\right) - \color{blue}{\sinh im}\right) \]

4. Applied rewrites 38.5%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(e^{-im} - \cosh im\right) - \sinh im\right)} \]

5. Taylor expanded in im around 0

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

7. Applied rewrites 99.6%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(-im\right)} - \sinh im\right) \]
8. Add Preprocessing

Alternative 18: 29.7% accurate, 105.7× speedup

Math

\[\begin{array}{l} \\ -im \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return -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 = -im
end function

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in im around 0

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in re around 0

   \[\leadsto -1 \cdot \color{blue}{im} \]
7. Step-by-step derivation

8. Applied rewrites 26.7%

   \[\leadsto -im \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025019 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (cos re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (cos re)) (- (exp (- 0 im)) (exp im)))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))