math.sin on complex, imaginary part

Percentage Accurate: 54.8% → 99.9%

Time: 10.7s

Alternatives: 19

Speedup: 2.5×

• 50.1% 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: 54.8% 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: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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 = sinh(-im) * cos(re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\cos re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

      \[\leadsto \cos re \cdot \left(\sinh \left(-im\right) \cdot \color{blue}{1}\right) \]

   8. metadata-eval N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lift-sinh.f64 N/A

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

   12. sinh-undef-rev N/A

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

   13. sinh-def N/A

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

   14. lift-sinh.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 87.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \sinh \left(-im\right)\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\left(\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))))
        (t_1 (sinh (- im))))
   (if (<= t_0 -1000000.0)
     t_1
     (if (<= t_0 0.01)
       (*
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0001984126984126984 (* im im)) 0.008333333333333333)
              im)
             im)
            0.16666666666666666)
           (* im im))
          1.0)
         im)
        (cos re))
       (* t_1 (fma -0.5 (* re re) 1.0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) - exp(im));
	double t_1 = sinh(-im);
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = ((((((((-0.0001984126984126984 * (im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * (im * im)) - 1.0) * im) * cos(re);
	} else {
		tmp = t_1 * fma(-0.5, (re * re), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = sinh(Float64(-im))
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * Float64(im * im)) - 1.0) * im) * cos(re));
	else
		tmp = Float64(t_1 * fma(-0.5, Float64(re * re), 1.0));
	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[Sinh[(-im)], $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], t$95$1, If[LessEqual[t$95$0, 0.01], N[(N[(N[(N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $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))) < -1e6

   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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 72.7

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

   5. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.7%

      \[\leadsto \sinh \left(-im\right) \]

   if -1e6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0100000000000000002

   1. Initial program 7.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\cos re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

         \[\leadsto \cos re \cdot \left(\sinh \left(-im\right) \cdot \color{blue}{1}\right) \]

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 99.8%

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

   if 0.0100000000000000002 < (*.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

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\cos re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

         \[\leadsto \cos re \cdot \left(\sinh \left(-im\right) \cdot \color{blue}{1}\right) \]

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 76.7

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

   9. Applied rewrites 76.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -1000000:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.01:\\ \;\;\;\;\left(\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(-im\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \sinh \left(-im\right)\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\left(\left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))))
        (t_1 (sinh (- im))))
   (if (<= t_0 -1000000.0)
     t_1
     (if (<= t_0 0.01)
       (*
        (*
         (-
          (*
           (* (- (* -0.008333333333333333 (* im im)) 0.16666666666666666) im)
           im)
          1.0)
         im)
        (cos re))
       (* t_1 (fma -0.5 (* re re) 1.0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) - exp(im));
	double t_1 = sinh(-im);
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = ((((((-0.008333333333333333 * (im * im)) - 0.16666666666666666) * im) * im) - 1.0) * im) * cos(re);
	} else {
		tmp = t_1 * fma(-0.5, (re * re), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = sinh(Float64(-im))
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * Float64(im * im)) - 0.16666666666666666) * im) * im) - 1.0) * im) * cos(re));
	else
		tmp = Float64(t_1 * fma(-0.5, Float64(re * re), 1.0));
	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[Sinh[(-im)], $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], t$95$1, If[LessEqual[t$95$0, 0.01], N[(N[(N[(N[(N[(N[(N[(-0.008333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $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))) < -1e6

   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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 72.7

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

   5. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.7%

      \[\leadsto \sinh \left(-im\right) \]

   if -1e6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0100000000000000002

   1. Initial program 7.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\cos re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

         \[\leadsto \cos re \cdot \left(\sinh \left(-im\right) \cdot \color{blue}{1}\right) \]

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.7

          \[\leadsto \left(\left(\left(\left(-0.008333333333333333 \cdot \color{blue}{\left(im \cdot im\right)} - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\right) \cdot \cos re \]

   9. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\right)} \cdot \cos re \]

   if 0.0100000000000000002 < (*.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

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\cos re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

         \[\leadsto \cos re \cdot \left(\sinh \left(-im\right) \cdot \color{blue}{1}\right) \]

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 76.7

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

   9. Applied rewrites 76.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -1000000:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.01:\\ \;\;\;\;\left(\left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(-im\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \sinh \left(-im\right)\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))))
        (t_1 (sinh (- im))))
   (if (<= t_0 -0.04)
     t_1
     (if (<= t_0 0.01)
       (* (* (fma (* -0.16666666666666666 im) im -1.0) im) (cos re))
       (* t_1 (fma -0.5 (* re re) 1.0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) - exp(im));
	double t_1 = sinh(-im);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = (fma((-0.16666666666666666 * im), im, -1.0) * im) * cos(re);
	} else {
		tmp = t_1 * fma(-0.5, (re * re), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = sinh(Float64(-im))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = Float64(Float64(fma(Float64(-0.16666666666666666 * im), im, -1.0) * im) * cos(re));
	else
		tmp = Float64(t_1 * fma(-0.5, Float64(re * re), 1.0));
	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[Sinh[(-im)], $MachinePrecision]}, If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, 0.01], N[(N[(N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision] * im), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $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))) < -0.0400000000000000008

   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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 73.1

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

   5. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.1%

      \[\leadsto \sinh \left(-im\right) \]

   if -0.0400000000000000008 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0100000000000000002

   1. Initial program 6.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   if 0.0100000000000000002 < (*.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

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\cos re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

         \[\leadsto \cos re \cdot \left(\sinh \left(-im\right) \cdot \color{blue}{1}\right) \]

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 76.7

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

   9. Applied rewrites 76.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.04:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(-im\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \sinh \left(-im\right)\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\left(\cos re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))))
        (t_1 (sinh (- im))))
   (if (<= t_0 -0.04)
     t_1
     (if (<= t_0 0.01)
       (* (* (cos re) im) (fma (* -0.16666666666666666 im) im -1.0))
       (* t_1 (fma -0.5 (* re re) 1.0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) - exp(im));
	double t_1 = sinh(-im);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = (cos(re) * im) * fma((-0.16666666666666666 * im), im, -1.0);
	} else {
		tmp = t_1 * fma(-0.5, (re * re), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = sinh(Float64(-im))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = Float64(Float64(cos(re) * im) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	else
		tmp = Float64(t_1 * fma(-0.5, Float64(re * re), 1.0));
	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[Sinh[(-im)], $MachinePrecision]}, If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, 0.01], N[(N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $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))) < -0.0400000000000000008

   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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 73.1

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

   5. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.1%

      \[\leadsto \sinh \left(-im\right) \]

   if -0.0400000000000000008 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0100000000000000002

   1. Initial program 6.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 99.6%

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

   if 0.0100000000000000002 < (*.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

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\cos re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

         \[\leadsto \cos re \cdot \left(\sinh \left(-im\right) \cdot \color{blue}{1}\right) \]

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 76.7

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

   9. Applied rewrites 76.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.04:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.01:\\ \;\;\;\;\left(\cos re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(-im\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \sinh \left(-im\right)\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))))
        (t_1 (sinh (- im))))
   (if (<= t_0 -0.04)
     t_1
     (if (<= t_0 0.01) (* (- (cos re)) im) (* t_1 (fma -0.5 (* re re) 1.0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) - exp(im));
	double t_1 = sinh(-im);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = -cos(re) * im;
	} else {
		tmp = t_1 * fma(-0.5, (re * re), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = sinh(Float64(-im))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = Float64(Float64(-cos(re)) * im);
	else
		tmp = Float64(t_1 * fma(-0.5, Float64(re * re), 1.0));
	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[Sinh[(-im)], $MachinePrecision]}, If[LessEqual[t$95$0, -0.04], t$95$1, If[LessEqual[t$95$0, 0.01], N[((-N[Cos[re], $MachinePrecision]) * im), $MachinePrecision], N[(t$95$1 * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $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))) < -0.0400000000000000008

   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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 73.1

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

   5. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.1%

      \[\leadsto \sinh \left(-im\right) \]

   if -0.0400000000000000008 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0100000000000000002

   1. Initial program 6.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.0100000000000000002 < (*.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

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\cos re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

         \[\leadsto \cos re \cdot \left(\sinh \left(-im\right) \cdot \color{blue}{1}\right) \]

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 76.7

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

   9. Applied rewrites 76.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.04:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.01:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(-im\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, re \cdot re, 1\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -0.04)
     (sinh (- im))
     (if (<= t_0 0.01)
       (* (- (cos re)) im)
       (*
        (*
         (fma
          (-
           (*
            (* (fma -0.001388888888888889 (* re re) 0.041666666666666664) re)
            re)
           0.5)
          (* re re)
          1.0)
         (*
          (-
           (*
            (-
             (*
              (*
               (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
               im)
              im)
             0.3333333333333333)
            (* im im))
           2.0)
          im))
        0.5)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -0.04) {
		tmp = sinh(-im);
	} else if (t_0 <= 0.01) {
		tmp = -cos(re) * im;
	} else {
		tmp = (fma((((fma(-0.001388888888888889, (re * re), 0.041666666666666664) * re) * re) - 0.5), (re * re), 1.0) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = sinh(Float64(-im));
	elseif (t_0 <= 0.01)
		tmp = Float64(Float64(-cos(re)) * im);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664) * re) * re) - 0.5), Float64(re * re), 1.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im)) * 0.5);
	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]}, If[LessEqual[t$95$0, -0.04], N[Sinh[(-im)], $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[((-N[Cos[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] * 0.5), $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))) < -0.0400000000000000008

   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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 73.1

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

   5. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.1%

      \[\leadsto \sinh \left(-im\right) \]

   if -0.0400000000000000008 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0100000000000000002

   1. Initial program 6.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.0100000000000000002 < (*.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

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 90.3%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) \cdot {re}^{2}} + 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, {re}^{2}, 1\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      4. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      7. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{720} \cdot {re}^{2} + \frac{1}{24}\right)} \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {re}^{2}, \frac{1}{24}\right)} \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      12. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{re \cdot re}, \frac{1}{24}\right) \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{re \cdot re}, \frac{1}{24}\right) \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{720}, re \cdot re, \frac{1}{24}\right) \cdot re\right) \cdot re - \frac{1}{2}, \color{blue}{re \cdot re}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      15. lower-*.f64 69.0

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, \color{blue}{re \cdot re}, 1\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot 0.5 \]

   10. Applied rewrites 69.0%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, re \cdot re, 1\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.04:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.01:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, re \cdot re, 1\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = sinh(-im);
	} else {
		tmp = (fma((((fma(-0.001388888888888889, (re * re), 0.041666666666666664) * re) * re) - 0.5), (re * re), 1.0) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)) * 0.5;
	}
	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 = sinh(Float64(-im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664) * re) * re) - 0.5), Float64(re * re), 1.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im)) * 0.5);
	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[Sinh[(-im)], $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] * 0.5), $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 40.5%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 30.1

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

   5. Applied rewrites 30.1%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.0%

      \[\leadsto \sinh \left(-im\right) \]

   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 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) \cdot {re}^{2}} + 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, {re}^{2}, 1\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      4. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      7. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{720} \cdot {re}^{2} + \frac{1}{24}\right)} \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {re}^{2}, \frac{1}{24}\right)} \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      12. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{re \cdot re}, \frac{1}{24}\right) \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{re \cdot re}, \frac{1}{24}\right) \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{720}, re \cdot re, \frac{1}{24}\right) \cdot re\right) \cdot re - \frac{1}{2}, \color{blue}{re \cdot re}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      15. lower-*.f64 68.1

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, \color{blue}{re \cdot re}, 1\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot 0.5 \]

   10. Applied rewrites 68.1%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, re \cdot re, 1\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

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

Alternative 9: 61.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, re \cdot re, 1\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (*
    (-
     (*
      (-
       (*
        (* (- (* -0.0001984126984126984 (* im im)) 0.008333333333333333) im)
        im)
       0.16666666666666666)
      (* im im))
     1.0)
    im)
   (*
    (*
     (fma
      (-
       (* (* (fma -0.001388888888888889 (* re re) 0.041666666666666664) re) re)
       0.5)
      (* re re)
      1.0)
     (*
      (-
       (*
        (-
         (*
          (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
          im)
         0.3333333333333333)
        (* im im))
       2.0)
      im))
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = (((((((-0.0001984126984126984 * (im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * (im * im)) - 1.0) * im;
	} else {
		tmp = (fma((((fma(-0.001388888888888889, (re * re), 0.041666666666666664) * re) * re) - 0.5), (re * re), 1.0) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im)) * 0.5;
	}
	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(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * Float64(im * im)) - 1.0) * im);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664) * re) * re) - 0.5), Float64(re * re), 1.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im)) * 0.5);
	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[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] * 0.5), $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 40.5%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 58.4%

      \[\leadsto \left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \color{blue}{im} \]

   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 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) \cdot {re}^{2}} + 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, {re}^{2}, 1\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      4. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      7. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{720} \cdot {re}^{2} + \frac{1}{24}\right)} \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {re}^{2}, \frac{1}{24}\right)} \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      12. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{re \cdot re}, \frac{1}{24}\right) \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{re \cdot re}, \frac{1}{24}\right) \cdot re\right) \cdot re - \frac{1}{2}, {re}^{2}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{720}, re \cdot re, \frac{1}{24}\right) \cdot re\right) \cdot re - \frac{1}{2}, \color{blue}{re \cdot re}, 1\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]

      15. lower-*.f64 68.1

          \[\leadsto \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, \color{blue}{re \cdot re}, 1\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot 0.5 \]

   10. Applied rewrites 68.1%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, re \cdot re, 1\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, re \cdot re, 1\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, re \cdot re, 1\right) \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (*
    (-
     (*
      (-
       (*
        (* (- (* -0.0001984126984126984 (* im im)) 0.008333333333333333) im)
        im)
       0.16666666666666666)
      (* im im))
     1.0)
    im)
   (*
    (*
     (fma
      (-
       (* (* (fma -0.001388888888888889 (* re re) 0.041666666666666664) re) re)
       0.5)
      (* re re)
      1.0)
     im)
    (fma (* -0.16666666666666666 im) im -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = (((((((-0.0001984126984126984 * (im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * (im * im)) - 1.0) * im;
	} else {
		tmp = (fma((((fma(-0.001388888888888889, (re * re), 0.041666666666666664) * re) * re) - 0.5), (re * re), 1.0) * im) * fma((-0.16666666666666666 * im), im, -1.0);
	}
	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(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * Float64(im * im)) - 1.0) * im);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664) * re) * re) - 0.5), Float64(re * re), 1.0) * im) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	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[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $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 40.5%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 58.4%

      \[\leadsto \left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \color{blue}{im} \]

   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 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 50.1%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot re\right) \cdot re - 0.5, re \cdot re, 1\right) \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.0005)
   (*
    (fma (* re re) -0.25 0.5)
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im) im)
      2.0)
     im))
   (*
    (-
     (*
      (-
       (*
        (* (- (* -0.0001984126984126984 (* im im)) 0.008333333333333333) im)
        im)
       0.16666666666666666)
      (* im im))
     1.0)
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.0005) {
		tmp = fma((re * re), -0.25, 0.5) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = (((((((-0.0001984126984126984 * (im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * (im * im)) - 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.0005)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * Float64(im * im)) - 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.0005], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -5.0000000000000001e-4

   1. Initial program 63.6%

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)} \cdot im\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)} - 2\right) \cdot im\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right) \cdot im} - 2\right) \cdot im\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right) \cdot im} - 2\right) \cdot im\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\color{blue}{\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right)} \cdot im - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\color{blue}{\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right)} \cdot im - 2\right) \cdot im\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)} \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      13. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      5. lower-*.f64 54.7

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

   8. Applied rewrites 54.7%

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

   if -5.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 55.8%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 79.7%

      \[\leadsto \left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 71.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot \left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.0005)
   (*
    (fma (* im im) -0.16666666666666666 -1.0)
    (* (fma -0.5 (* re re) 1.0) im))
   (*
    (-
     (*
      (-
       (*
        (* (- (* -0.0001984126984126984 (* im im)) 0.008333333333333333) im)
        im)
       0.16666666666666666)
      (* im im))
     1.0)
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.0005) {
		tmp = fma((im * im), -0.16666666666666666, -1.0) * (fma(-0.5, (re * re), 1.0) * im);
	} else {
		tmp = (((((((-0.0001984126984126984 * (im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * (im * im)) - 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.0005)
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, -1.0) * Float64(fma(-0.5, Float64(re * re), 1.0) * im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * Float64(im * im)) - 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.0005], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -5.0000000000000001e-4

   1. Initial program 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 76.7%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 53.2%

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

   if -5.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 55.8%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 79.7%

      \[\leadsto \left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot \left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot \left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.0005)
   (*
    (fma (* im im) -0.16666666666666666 -1.0)
    (* (fma -0.5 (* re re) 1.0) im))
   (*
    (-
     (* (* (- (* -0.008333333333333333 (* im im)) 0.16666666666666666) im) im)
     1.0)
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.0005) {
		tmp = fma((im * im), -0.16666666666666666, -1.0) * (fma(-0.5, (re * re), 1.0) * im);
	} else {
		tmp = (((((-0.008333333333333333 * (im * im)) - 0.16666666666666666) * im) * im) - 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.0005)
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, -1.0) * Float64(fma(-0.5, Float64(re * re), 1.0) * im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * Float64(im * im)) - 0.16666666666666666) * im) * im) - 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.0005], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.008333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -5.0000000000000001e-4

   1. Initial program 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 76.7%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 53.2%

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

   if -5.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 55.8%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 74.8%

      \[\leadsto \left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot \left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(0.5 \cdot re\right), re, -im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.0005)
   (fma (* im (* 0.5 re)) re (- im))
   (*
    (-
     (* (* (- (* -0.008333333333333333 (* im im)) 0.16666666666666666) im) im)
     1.0)
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.0005) {
		tmp = fma((im * (0.5 * re)), re, -im);
	} else {
		tmp = (((((-0.008333333333333333 * (im * im)) - 0.16666666666666666) * im) * im) - 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.0005)
		tmp = fma(Float64(im * Float64(0.5 * re)), re, Float64(-im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * Float64(im * im)) - 0.16666666666666666) * im) * im) - 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.0005], N[(N[(im * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] * re + (-im)), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.008333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -5.0000000000000001e-4

   1. Initial program 63.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 43.3

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.8%

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

   10. Applied rewrites 45.9%

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

   if -5.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 55.8%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 74.8%

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

Alternative 15: 63.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(0.5 \cdot re\right), re, -im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.0005)
   (fma (* im (* 0.5 re)) re (- im))
   (* (fma (* im im) -0.16666666666666666 -1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.0005) {
		tmp = fma((im * (0.5 * re)), re, -im);
	} else {
		tmp = fma((im * im), -0.16666666666666666, -1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.0005)
		tmp = fma(Float64(im * Float64(0.5 * re)), re, Float64(-im));
	else
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, -1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.0005], N[(N[(im * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] * re + (-im)), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -5.0000000000000001e-4

   1. Initial program 63.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 43.3

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.8%

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

   10. Applied rewrites 45.9%

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

   if -5.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 66.7%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(0.5 \cdot re\right), re, -im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5 \cdot re, re, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.0005)
   (* im (fma (* 0.5 re) re -1.0))
   (* (fma (* im im) -0.16666666666666666 -1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.0005) {
		tmp = im * fma((0.5 * re), re, -1.0);
	} else {
		tmp = fma((im * im), -0.16666666666666666, -1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.0005)
		tmp = Float64(im * fma(Float64(0.5 * re), re, -1.0));
	else
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, -1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.0005], N[(im * N[(N[(0.5 * re), $MachinePrecision] * re + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -5.0000000000000001e-4

   1. Initial program 63.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 43.3

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.8%

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

   if -5.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 66.7%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5 \cdot re, re, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 39.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5 \cdot re, re, -1\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.0005) (* im (fma (* 0.5 re) re -1.0)) (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.0005) {
		tmp = im * fma((0.5 * re), re, -1.0);
	} else {
		tmp = -im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.0005)
		tmp = Float64(im * fma(Float64(0.5 * re), re, -1.0));
	else
		tmp = Float64(-im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.0005], N[(im * N[(N[(0.5 * re), $MachinePrecision] * re + -1.0), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -5.0000000000000001e-4

   1. Initial program 63.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 43.3

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.8%

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

   if -5.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 55.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.6%

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

Alternative 18: 39.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0005:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.0005) (* im (* (* re re) 0.5)) (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.0005) {
		tmp = im * ((re * re) * 0.5);
	} else {
		tmp = -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)) <= (-0.0005d0)) then
        tmp = im * ((re * re) * 0.5d0)
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.cos(re)) <= -0.0005) {
		tmp = im * ((re * re) * 0.5);
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.cos(re)) <= -0.0005:
		tmp = im * ((re * re) * 0.5)
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.0005)
		tmp = Float64(im * Float64(Float64(re * re) * 0.5));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * cos(re)) <= -0.0005)
		tmp = im * ((re * re) * 0.5);
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.0005], N[(im * N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -5.0000000000000001e-4

   1. Initial program 63.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 43.3

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

   5. Applied rewrites 43.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.8%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 45.8%

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

   if -5.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 55.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.6%

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

Alternative 19: 29.5% 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 57.7%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-cos.f64 48.1

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

5. Applied rewrites 48.1%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 27.3%

   \[\leadsto -im \]
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 2024358 
(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))))