Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5

Percentage Accurate: 99.3% → 99.4%

Time: 10.4s

Alternatives: 33

Speedup: 1.1×

Specification

Math

\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (*
   3.0
   (+
    (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
    (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))

C

double code(double x, double y) {
	return (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
}

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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) - cos(y)))) / (3.0d0 * ((1.0d0 + (((sqrt(5.0d0) - 1.0d0) / 2.0d0) * cos(x))) + (((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) - Math.cos(y)))) / (3.0 * ((1.0 + (((Math.sqrt(5.0) - 1.0) / 2.0) * Math.cos(x))) + (((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y))));
}

Python

def code(x, y):
	return (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) - math.cos(y)))) / (3.0 * ((1.0 + (((math.sqrt(5.0) - 1.0) / 2.0) * math.cos(x))) + (((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.3% accurate, 1.0× speedup

Math

\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (*
   3.0
   (+
    (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
    (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))

C

double code(double x, double y) {
	return (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
}

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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) - cos(y)))) / (3.0d0 * ((1.0d0 + (((sqrt(5.0d0) - 1.0d0) / 2.0d0) * cos(x))) + (((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) - Math.cos(y)))) / (3.0 * ((1.0 + (((Math.sqrt(5.0) - 1.0) / 2.0) * Math.cos(x))) + (((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y))));
}

Python

def code(x, y):
	return (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) - math.cos(y)))) / (3.0 * ((1.0 + (((math.sqrt(5.0) - 1.0) / 2.0) * math.cos(x))) + (((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (- (cos x) (cos y)) (* (sqrt 2.0) (fma (sin x) -0.0625 (sin y))))
   (fma (sin y) -0.0625 (sin x))
   2.0)
  (fma
   3.0
   (* (fma (- (sqrt 5.0) 1.0) (cos x) (* 0.7639320225002103 (cos y))) 0.5)
   3.0)))

C

double code(double x, double y) {
	return fma(((cos(x) - cos(y)) * (sqrt(2.0) * fma(sin(x), -0.0625, sin(y)))), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(3.0, (fma((sqrt(5.0) - 1.0), cos(x), (0.7639320225002103 * cos(y))) * 0.5), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * fma(sin(x), -0.0625, sin(y)))), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(3.0, Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(0.7639320225002103 * cos(y))) * 0.5), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

4. Evaluated real constant 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   5. lower-*.f64 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   8. lower-fma.f64 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right)} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

   5. lift-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \sin x + \sin y\right)}\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\sin x \cdot \frac{-1}{16}} + \sin y\right)\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

   7. lift-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)}\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

   8. lower-*.f64 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

8. Applied rewrites 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
9. Add Preprocessing

Alternative 2: 99.4% accurate, 1.1× speedup

Math

\[\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(1.2360679774997898, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (* (- (cos x) (cos y)) (sqrt 2.0)) (fma -0.0625 (sin x) (sin y)))
   (fma (sin y) -0.0625 (sin x))
   2.0)
  (fma
   3.0
   (* (fma 1.2360679774997898 (cos x) (* 0.7639320225002103 (cos y))) 0.5)
   3.0)))

C

double code(double x, double y) {
	return fma((((cos(x) - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(x), sin(y))), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(3.0, (fma(1.2360679774997898, cos(x), (0.7639320225002103 * cos(y))) * 0.5), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(x), sin(y))), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(3.0, Float64(fma(1.2360679774997898, cos(x), Float64(0.7639320225002103 * cos(y))) * 0.5), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.2360679774997898 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

4. Evaluated real constant 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   5. lower-*.f64 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

   8. lower-fma.f64 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

7. Evaluated real constant 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\color{blue}{1.2360679774997898}, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
8. Add Preprocessing

Alternative 3: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ t_3 := \sin x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -3:\\ \;\;\;\;\frac{2 + \left(t\_3 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \cos y \cdot \left(-0.5 \cdot t\_0\right)\right)}\\ \mathbf{elif}\;x \leq 0.023:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot \sqrt{2}, \left(\sin y + x \cdot \left(0.010416666666666666 \cdot {x}^{2} - 0.0625\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_2, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y - \cos x, t\_3 \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), -2\right) \cdot \frac{1}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 1.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (* (sin x) (sqrt 2.0))))
   (if (<= x -3.0)
     (/
      (+ 2.0 (* (* t_3 (- (sin y) (/ (sin x) 16.0))) t_1))
      (*
       3.0
       (fma
        (fma (sqrt 5.0) 0.5 -0.5)
        (cos x)
        (- 1.0 (* (cos y) (* -0.5 t_0))))))
     (if (<= x 0.023)
       (/
        (fma
         (* t_1 (sqrt 2.0))
         (*
          (+ (sin y) (* x (- (* 0.010416666666666666 (pow x 2.0)) 0.0625)))
          (fma (sin y) -0.0625 (sin x)))
         2.0)
        (+
         3.0
         (* 3.0 (/ (fma t_2 (cos x) (* 0.7639320225002103 (cos y))) 2.0))))
       (*
        (fma (- (cos y) (cos x)) (* t_3 (fma -0.0625 (sin x) (sin y))) -2.0)
        (/ 1.0 (- (fma (fma t_2 (cos x) (* t_0 (cos y))) 1.5 3.0))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = sin(x) * sqrt(2.0);
	double tmp;
	if (x <= -3.0) {
		tmp = (2.0 + ((t_3 * (sin(y) - (sin(x) / 16.0))) * t_1)) / (3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), (1.0 - (cos(y) * (-0.5 * t_0)))));
	} else if (x <= 0.023) {
		tmp = fma((t_1 * sqrt(2.0)), ((sin(y) + (x * ((0.010416666666666666 * pow(x, 2.0)) - 0.0625))) * fma(sin(y), -0.0625, sin(x))), 2.0) / (3.0 + (3.0 * (fma(t_2, cos(x), (0.7639320225002103 * cos(y))) / 2.0)));
	} else {
		tmp = fma((cos(y) - cos(x)), (t_3 * fma(-0.0625, sin(x), sin(y))), -2.0) * (1.0 / -fma(fma(t_2, cos(x), (t_0 * cos(y))), 1.5, 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(sin(x) * sqrt(2.0))
	tmp = 0.0
	if (x <= -3.0)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_3 * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(1.0 - Float64(cos(y) * Float64(-0.5 * t_0))))));
	elseif (x <= 0.023)
		tmp = Float64(fma(Float64(t_1 * sqrt(2.0)), Float64(Float64(sin(y) + Float64(x * Float64(Float64(0.010416666666666666 * (x ^ 2.0)) - 0.0625))) * fma(sin(y), -0.0625, sin(x))), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_2, cos(x), Float64(0.7639320225002103 * cos(y))) / 2.0))));
	else
		tmp = Float64(fma(Float64(cos(y) - cos(x)), Float64(t_3 * fma(-0.0625, sin(x), sin(y))), -2.0) * Float64(1.0 / Float64(-fma(fma(t_2, cos(x), Float64(t_0 * cos(y))), 1.5, 3.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.0], N[(N[(2.0 + N[(N[(t$95$3 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(1.0 - N[(N[Cos[y], $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.023], N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(x * N[(N[(0.010416666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * N[(1.0 / (-N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \cos y \cdot \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   if -3 < x < 0.023

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\left(\sin y + x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)\right)} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y + \color{blue}{x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)}\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y + \color{blue}{x} \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y + x \cdot \color{blue}{\left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)}\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y + x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \color{blue}{\frac{1}{16}}\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y + x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      6. lower-pow.f64 51.1%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y + x \cdot \left(0.010416666666666666 \cdot {x}^{2} - 0.0625\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 51.1%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\left(\sin y + x \cdot \left(0.010416666666666666 \cdot {x}^{2} - 0.0625\right)\right)} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if 0.023 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y - \cos x, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), -2\right) \cdot \frac{1}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1.5, 3\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ t_3 := \sin x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -3:\\ \;\;\;\;\frac{2 + \left(t\_3 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \cos y \cdot \left(-0.5 \cdot t\_0\right)\right)}\\ \mathbf{elif}\;x \leq 0.023:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_1 \cdot \sqrt{2}\right) \cdot \left(\sin y + x \cdot \left(0.010416666666666666 \cdot {x}^{2} - 0.0625\right)\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_2, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y - \cos x, t\_3 \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), -2\right) \cdot \frac{1}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 1.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (* (sin x) (sqrt 2.0))))
   (if (<= x -3.0)
     (/
      (+ 2.0 (* (* t_3 (- (sin y) (/ (sin x) 16.0))) t_1))
      (*
       3.0
       (fma
        (fma (sqrt 5.0) 0.5 -0.5)
        (cos x)
        (- 1.0 (* (cos y) (* -0.5 t_0))))))
     (if (<= x 0.023)
       (/
        (fma
         (*
          (* t_1 (sqrt 2.0))
          (+ (sin y) (* x (- (* 0.010416666666666666 (pow x 2.0)) 0.0625))))
         (fma (sin y) -0.0625 (sin x))
         2.0)
        (fma 3.0 (* (fma t_2 (cos x) (* 0.7639320225002103 (cos y))) 0.5) 3.0))
       (*
        (fma (- (cos y) (cos x)) (* t_3 (fma -0.0625 (sin x) (sin y))) -2.0)
        (/ 1.0 (- (fma (fma t_2 (cos x) (* t_0 (cos y))) 1.5 3.0))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = sin(x) * sqrt(2.0);
	double tmp;
	if (x <= -3.0) {
		tmp = (2.0 + ((t_3 * (sin(y) - (sin(x) / 16.0))) * t_1)) / (3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), (1.0 - (cos(y) * (-0.5 * t_0)))));
	} else if (x <= 0.023) {
		tmp = fma(((t_1 * sqrt(2.0)) * (sin(y) + (x * ((0.010416666666666666 * pow(x, 2.0)) - 0.0625)))), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(3.0, (fma(t_2, cos(x), (0.7639320225002103 * cos(y))) * 0.5), 3.0);
	} else {
		tmp = fma((cos(y) - cos(x)), (t_3 * fma(-0.0625, sin(x), sin(y))), -2.0) * (1.0 / -fma(fma(t_2, cos(x), (t_0 * cos(y))), 1.5, 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(sin(x) * sqrt(2.0))
	tmp = 0.0
	if (x <= -3.0)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_3 * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(1.0 - Float64(cos(y) * Float64(-0.5 * t_0))))));
	elseif (x <= 0.023)
		tmp = Float64(fma(Float64(Float64(t_1 * sqrt(2.0)) * Float64(sin(y) + Float64(x * Float64(Float64(0.010416666666666666 * (x ^ 2.0)) - 0.0625)))), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(3.0, Float64(fma(t_2, cos(x), Float64(0.7639320225002103 * cos(y))) * 0.5), 3.0));
	else
		tmp = Float64(fma(Float64(cos(y) - cos(x)), Float64(t_3 * fma(-0.0625, sin(x), sin(y))), -2.0) * Float64(1.0 / Float64(-fma(fma(t_2, cos(x), Float64(t_0 * cos(y))), 1.5, 3.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.0], N[(N[(2.0 + N[(N[(t$95$3 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(1.0 - N[(N[Cos[y], $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.023], N[(N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(x * N[(N[(0.010416666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * N[(1.0 / (-N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \cos y \cdot \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   if -3 < x < 0.023

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin y + x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y + \color{blue}{x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)}\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y + \color{blue}{x} \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y + x \cdot \color{blue}{\left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)}\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y + x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \color{blue}{\frac{1}{16}}\right)\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y + x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      6. lower-pow.f64 51.1%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y + x \cdot \left(0.010416666666666666 \cdot {x}^{2} - 0.0625\right)\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 51.1%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin y + x \cdot \left(0.010416666666666666 \cdot {x}^{2} - 0.0625\right)\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   if 0.023 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y - \cos x, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), -2\right) \cdot \frac{1}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1.5, 3\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_3 := \sin x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -2.4:\\ \;\;\;\;\frac{2 + \left(t\_3 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \cos y \cdot \left(-0.5 \cdot t\_0\right)\right)}\\ \mathbf{elif}\;x \leq 0.023:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_2 - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - t\_2 \cdot 0.0625\right) \cdot t\_1, 2\right) \cdot \frac{1}{\mathsf{fma}\left(t\_0 \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y - \cos x, t\_3 \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), -2\right) \cdot \frac{1}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0 \cdot \cos y\right), 1.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_3 (* (sin x) (sqrt 2.0))))
   (if (<= x -2.4)
     (/
      (+ 2.0 (* (* t_3 (- (sin y) (/ (sin x) 16.0))) t_1))
      (*
       3.0
       (fma
        (fma (sqrt 5.0) 0.5 -0.5)
        (cos x)
        (- 1.0 (* (cos y) (* -0.5 t_0))))))
     (if (<= x 0.023)
       (*
        (fma
         (* (- t_2 (* (sin y) 0.0625)) (sqrt 2.0))
         (* (- (sin y) (* t_2 0.0625)) t_1)
         2.0)
        (/
         1.0
         (*
          (fma (* t_0 0.5) (cos y) (fma (cos x) 0.6180339887498949 1.0))
          3.0)))
       (*
        (fma (- (cos y) (cos x)) (* t_3 (fma -0.0625 (sin x) (sin y))) -2.0)
        (/
         1.0
         (-
          (fma (fma (- (sqrt 5.0) 1.0) (cos x) (* t_0 (cos y))) 1.5 3.0))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_3 = sin(x) * sqrt(2.0);
	double tmp;
	if (x <= -2.4) {
		tmp = (2.0 + ((t_3 * (sin(y) - (sin(x) / 16.0))) * t_1)) / (3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), (1.0 - (cos(y) * (-0.5 * t_0)))));
	} else if (x <= 0.023) {
		tmp = fma(((t_2 - (sin(y) * 0.0625)) * sqrt(2.0)), ((sin(y) - (t_2 * 0.0625)) * t_1), 2.0) * (1.0 / (fma((t_0 * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0)) * 3.0));
	} else {
		tmp = fma((cos(y) - cos(x)), (t_3 * fma(-0.0625, sin(x), sin(y))), -2.0) * (1.0 / -fma(fma((sqrt(5.0) - 1.0), cos(x), (t_0 * cos(y))), 1.5, 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_3 = Float64(sin(x) * sqrt(2.0))
	tmp = 0.0
	if (x <= -2.4)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_3 * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(1.0 - Float64(cos(y) * Float64(-0.5 * t_0))))));
	elseif (x <= 0.023)
		tmp = Float64(fma(Float64(Float64(t_2 - Float64(sin(y) * 0.0625)) * sqrt(2.0)), Float64(Float64(sin(y) - Float64(t_2 * 0.0625)) * t_1), 2.0) * Float64(1.0 / Float64(fma(Float64(t_0 * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0)) * 3.0)));
	else
		tmp = Float64(fma(Float64(cos(y) - cos(x)), Float64(t_3 * fma(-0.0625, sin(x), sin(y))), -2.0) * Float64(1.0 / Float64(-fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_0 * cos(y))), 1.5, 3.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4], N[(N[(2.0 + N[(N[(t$95$3 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(1.0 - N[(N[Cos[y], $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.023], N[(N[(N[(N[(t$95$2 - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(t$95$2 * 0.0625), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / N[(N[(N[(t$95$0 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * 0.6180339887498949 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * N[(1.0 / (-N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.39999999999999991

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \cos y \cdot \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   if -2.39999999999999991 < x < 0.023

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.1%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 51.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 51.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 51.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right) \cdot 3}} \]

   if 0.023 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y - \cos x, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), -2\right) \cdot \frac{1}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1.5, 3\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sin x \cdot \sqrt{2}\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_2 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ t_3 := 3 - \sqrt{5}\\ t_4 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_3 \cdot \cos y\right), 1.5, 3\right)\\ t_5 := \cos x - \cos y\\ \mathbf{if}\;x \leq -2.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot t\_5, t\_0, 2\right)}{t\_4}\\ \mathbf{elif}\;x \leq 0.023:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_1 - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - t\_1 \cdot 0.0625\right) \cdot t\_5, 2\right) \cdot \frac{1}{\mathsf{fma}\left(t\_3 \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos y - \cos x, t\_0 \cdot t\_2, -2\right) \cdot \frac{1}{-t\_4}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (sqrt 2.0)))
        (t_1 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_2 (fma -0.0625 (sin x) (sin y)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (fma (fma (- (sqrt 5.0) 1.0) (cos x) (* t_3 (cos y))) 1.5 3.0))
        (t_5 (- (cos x) (cos y))))
   (if (<= x -2.4)
     (/ (fma (* t_2 t_5) t_0 2.0) t_4)
     (if (<= x 0.023)
       (*
        (fma
         (* (- t_1 (* (sin y) 0.0625)) (sqrt 2.0))
         (* (- (sin y) (* t_1 0.0625)) t_5)
         2.0)
        (/
         1.0
         (*
          (fma (* t_3 0.5) (cos y) (fma (cos x) 0.6180339887498949 1.0))
          3.0)))
       (* (fma (- (cos y) (cos x)) (* t_0 t_2) -2.0) (/ 1.0 (- t_4)))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * sqrt(2.0);
	double t_1 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_2 = fma(-0.0625, sin(x), sin(y));
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = fma(fma((sqrt(5.0) - 1.0), cos(x), (t_3 * cos(y))), 1.5, 3.0);
	double t_5 = cos(x) - cos(y);
	double tmp;
	if (x <= -2.4) {
		tmp = fma((t_2 * t_5), t_0, 2.0) / t_4;
	} else if (x <= 0.023) {
		tmp = fma(((t_1 - (sin(y) * 0.0625)) * sqrt(2.0)), ((sin(y) - (t_1 * 0.0625)) * t_5), 2.0) * (1.0 / (fma((t_3 * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0)) * 3.0));
	} else {
		tmp = fma((cos(y) - cos(x)), (t_0 * t_2), -2.0) * (1.0 / -t_4);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * sqrt(2.0))
	t_1 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_2 = fma(-0.0625, sin(x), sin(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_3 * cos(y))), 1.5, 3.0)
	t_5 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -2.4)
		tmp = Float64(fma(Float64(t_2 * t_5), t_0, 2.0) / t_4);
	elseif (x <= 0.023)
		tmp = Float64(fma(Float64(Float64(t_1 - Float64(sin(y) * 0.0625)) * sqrt(2.0)), Float64(Float64(sin(y) - Float64(t_1 * 0.0625)) * t_5), 2.0) * Float64(1.0 / Float64(fma(Float64(t_3 * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0)) * 3.0)));
	else
		tmp = Float64(fma(Float64(cos(y) - cos(x)), Float64(t_0 * t_2), -2.0) * Float64(1.0 / Float64(-t_4)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4], N[(N[(N[(t$95$2 * t$95$5), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[x, 0.023], N[(N[(N[(N[(t$95$1 - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(t$95$1 * 0.0625), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / N[(N[(N[(t$95$3 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * 0.6180339887498949 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$2), $MachinePrecision] + -2.0), $MachinePrecision] * N[(1.0 / (-t$95$4)), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.39999999999999991

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), \sin x \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1.5, 3\right)}} \]

   if -2.39999999999999991 < x < 0.023

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.1%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 51.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 51.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 51.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right) \cdot 3}} \]

   if 0.023 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y - \cos x, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), -2\right) \cdot \frac{1}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1.5, 3\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos x - \cos y\\ t_3 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_2, \sin x \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1 \cdot \cos y\right), 1.5, 3\right)}\\ \mathbf{if}\;x \leq -2.4:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.023:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - t\_0 \cdot 0.0625\right) \cdot t\_2, 2\right) \cdot \frac{1}{\mathsf{fma}\left(t\_1 \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (cos x) (cos y)))
        (t_3
         (/
          (fma
           (* (fma -0.0625 (sin x) (sin y)) t_2)
           (* (sin x) (sqrt 2.0))
           2.0)
          (fma (fma (- (sqrt 5.0) 1.0) (cos x) (* t_1 (cos y))) 1.5 3.0))))
   (if (<= x -2.4)
     t_3
     (if (<= x 0.023)
       (*
        (fma
         (* (- t_0 (* (sin y) 0.0625)) (sqrt 2.0))
         (* (- (sin y) (* t_0 0.0625)) t_2)
         2.0)
        (/
         1.0
         (*
          (fma (* t_1 0.5) (cos y) (fma (cos x) 0.6180339887498949 1.0))
          3.0)))
       t_3))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = fma((fma(-0.0625, sin(x), sin(y)) * t_2), (sin(x) * sqrt(2.0)), 2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), (t_1 * cos(y))), 1.5, 3.0);
	double tmp;
	if (x <= -2.4) {
		tmp = t_3;
	} else if (x <= 0.023) {
		tmp = fma(((t_0 - (sin(y) * 0.0625)) * sqrt(2.0)), ((sin(y) - (t_0 * 0.0625)) * t_2), 2.0) * (1.0 / (fma((t_1 * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0)) * 3.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(fma(Float64(fma(-0.0625, sin(x), sin(y)) * t_2), Float64(sin(x) * sqrt(2.0)), 2.0) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_1 * cos(y))), 1.5, 3.0))
	tmp = 0.0
	if (x <= -2.4)
		tmp = t_3;
	elseif (x <= 0.023)
		tmp = Float64(fma(Float64(Float64(t_0 - Float64(sin(y) * 0.0625)) * sqrt(2.0)), Float64(Float64(sin(y) - Float64(t_0 * 0.0625)) * t_2), 2.0) * Float64(1.0 / Float64(fma(Float64(t_1 * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0)) * 3.0)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4], t$95$3, If[LessEqual[x, 0.023], N[(N[(N[(N[(t$95$0 - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(t$95$0 * 0.0625), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / N[(N[(N[(t$95$1 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * 0.6180339887498949 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999991 or 0.023 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), \sin x \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1.5, 3\right)}} \]

   if -2.39999999999999991 < x < 0.023

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.1%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 51.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 51.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 51.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right) \cdot 3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right)\\ t_3 := \left(\cos x - \cos y\right) \cdot \sqrt{2}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3 \cdot \sin y, t\_0, 2\right)}{\mathsf{fma}\left(3, t\_2 \cdot 0.5, 3\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot t\_0, 2\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, \sin y \cdot t\_0, 2\right)}{3 + 3 \cdot \frac{t\_2}{2}}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (fma t_1 (cos x) (* 0.7639320225002103 (cos y))))
        (t_3 (* (- (cos x) (cos y)) (sqrt 2.0))))
   (if (<= y -4.8e-7)
     (/ (fma (* t_3 (sin y)) t_0 2.0) (fma 3.0 (* t_2 0.5) 3.0))
     (if (<= y 4.2e-10)
       (/
        (fma t_3 (* (fma (sin x) -0.0625 (sin y)) t_0) 2.0)
        (+ 3.0 (* 1.5 (+ 0.7639320225002103 (* (cos x) t_1)))))
       (/ (fma t_3 (* (sin y) t_0) 2.0) (+ 3.0 (* 3.0 (/ t_2 2.0))))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma(t_1, cos(x), (0.7639320225002103 * cos(y)));
	double t_3 = (cos(x) - cos(y)) * sqrt(2.0);
	double tmp;
	if (y <= -4.8e-7) {
		tmp = fma((t_3 * sin(y)), t_0, 2.0) / fma(3.0, (t_2 * 0.5), 3.0);
	} else if (y <= 4.2e-10) {
		tmp = fma(t_3, (fma(sin(x), -0.0625, sin(y)) * t_0), 2.0) / (3.0 + (1.5 * (0.7639320225002103 + (cos(x) * t_1))));
	} else {
		tmp = fma(t_3, (sin(y) * t_0), 2.0) / (3.0 + (3.0 * (t_2 / 2.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y)))
	t_3 = Float64(Float64(cos(x) - cos(y)) * sqrt(2.0))
	tmp = 0.0
	if (y <= -4.8e-7)
		tmp = Float64(fma(Float64(t_3 * sin(y)), t_0, 2.0) / fma(3.0, Float64(t_2 * 0.5), 3.0));
	elseif (y <= 4.2e-10)
		tmp = Float64(fma(t_3, Float64(fma(sin(x), -0.0625, sin(y)) * t_0), 2.0) / Float64(3.0 + Float64(1.5 * Float64(0.7639320225002103 + Float64(cos(x) * t_1)))));
	else
		tmp = Float64(fma(t_3, Float64(sin(y) * t_0), 2.0) / Float64(3.0 + Float64(3.0 * Float64(t_2 / 2.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e-7], N[(N[(N[(t$95$3 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$2 * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e-10], N[(N[(t$95$3 * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(0.7639320225002103 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 * N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.79999999999999957e-7

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sin y}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-sin.f64 64.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 64.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sin y}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   if -4.79999999999999957e-7 < y < 4.2e-10

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + \color{blue}{\frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \color{blue}{\left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)} \]

      6. lower-sqrt.f64 60.7%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 60.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + \color{blue}{1.5 \cdot \left(0.7639320225002103 + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   if 4.2e-10 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\sin y} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-sin.f64 64.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \sin y \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 64.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\sin y} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(\cos x - \cos y\right) \cdot \sqrt{2}\\ t_3 := \frac{\mathsf{fma}\left(t\_2 \cdot \sin y, t\_0, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot t\_0, 2\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (- (cos x) (cos y)) (sqrt 2.0)))
        (t_3
         (/
          (fma (* t_2 (sin y)) t_0 2.0)
          (fma
           3.0
           (* (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 0.5)
           3.0))))
   (if (<= y -4.8e-7)
     t_3
     (if (<= y 4.2e-10)
       (/
        (fma t_2 (* (fma (sin x) -0.0625 (sin y)) t_0) 2.0)
        (+ 3.0 (* 1.5 (+ 0.7639320225002103 (* (cos x) t_1)))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (cos(x) - cos(y)) * sqrt(2.0);
	double t_3 = fma((t_2 * sin(y)), t_0, 2.0) / fma(3.0, (fma(t_1, cos(x), (0.7639320225002103 * cos(y))) * 0.5), 3.0);
	double tmp;
	if (y <= -4.8e-7) {
		tmp = t_3;
	} else if (y <= 4.2e-10) {
		tmp = fma(t_2, (fma(sin(x), -0.0625, sin(y)) * t_0), 2.0) / (3.0 + (1.5 * (0.7639320225002103 + (cos(x) * t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(cos(x) - cos(y)) * sqrt(2.0))
	t_3 = Float64(fma(Float64(t_2 * sin(y)), t_0, 2.0) / fma(3.0, Float64(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))) * 0.5), 3.0))
	tmp = 0.0
	if (y <= -4.8e-7)
		tmp = t_3;
	elseif (y <= 4.2e-10)
		tmp = Float64(fma(t_2, Float64(fma(sin(x), -0.0625, sin(y)) * t_0), 2.0) / Float64(3.0 + Float64(1.5 * Float64(0.7639320225002103 + Float64(cos(x) * t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e-7], t$95$3, If[LessEqual[y, 4.2e-10], N[(N[(t$95$2 * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(0.7639320225002103 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.79999999999999957e-7 or 4.2e-10 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sin y}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-sin.f64 64.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 64.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sin y}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   if -4.79999999999999957e-7 < y < 4.2e-10

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + \color{blue}{\frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \color{blue}{\left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)} \]

      6. lower-sqrt.f64 60.7%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 60.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + \color{blue}{1.5 \cdot \left(0.7639320225002103 + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\mathsf{fma}\left(\left(t\_1 \cdot \sqrt{2}\right) \cdot \sin y, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_0, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}\\ \mathbf{if}\;y \leq -1.42 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (/
          (fma
           (* (* t_1 (sqrt 2.0)) (sin y))
           (fma (sin y) -0.0625 (sin x))
           2.0)
          (fma
           3.0
           (* (fma t_0 (cos x) (* 0.7639320225002103 (cos y))) 0.5)
           3.0))))
   (if (<= y -1.42e-6)
     t_2
     (if (<= y 4.2e-10)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_1))
        (*
         3.0
         (+
          (+ 1.0 (* (/ t_0 2.0) (cos x)))
          (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = fma(((t_1 * sqrt(2.0)) * sin(y)), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(3.0, (fma(t_0, cos(x), (0.7639320225002103 * cos(y))) * 0.5), 3.0);
	double tmp;
	if (y <= -1.42e-6) {
		tmp = t_2;
	} else if (y <= 4.2e-10) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_1)) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(fma(Float64(Float64(t_1 * sqrt(2.0)) * sin(y)), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(3.0, Float64(fma(t_0, cos(x), Float64(0.7639320225002103 * cos(y))) * 0.5), 3.0))
	tmp = 0.0
	if (y <= -1.42e-6)
		tmp = t_2;
	elseif (y <= 4.2e-10)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.42e-6], t$95$2, If[LessEqual[y, 4.2e-10], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.42e-6 or 4.2e-10 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sin y}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-sin.f64 64.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 64.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sin y}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   if -1.42e-6 < y < 4.2e-10

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\color{blue}{y}}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\color{blue}{y}}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(\color{blue}{y} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(\color{blue}{y} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.07:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - t\_0 \cdot 0.0625\right) \cdot t\_2, 2\right) \cdot \frac{1}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - t\_1 \cdot \cos x, -0.5, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y))))
   (if (<= x -0.07)
     (/
      (fma (* t_2 (sqrt 2.0)) (* -0.0625 (pow (sin x) 2.0)) 2.0)
      (+ 3.0 (* 3.0 (/ (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 2.0))))
     (if (<= x 0.039)
       (*
        (fma
         (* (- t_0 (* (sin y) 0.0625)) (sqrt 2.0))
         (* (- (sin y) (* t_0 0.0625)) t_2)
         2.0)
        (/
         1.0
         (*
          (fma
           (* (- 3.0 (sqrt 5.0)) 0.5)
           (cos y)
           (fma (cos x) 0.6180339887498949 1.0))
          3.0)))
       (/
        (+
         2.0
         (*
          (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) 1.0)))
        (*
         3.0
         (fma
          (- (* (- (sqrt 5.0) 3.0) (cos y)) (* t_1 (cos x)))
          -0.5
          1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.07) {
		tmp = fma((t_2 * sqrt(2.0)), (-0.0625 * pow(sin(x), 2.0)), 2.0) / (3.0 + (3.0 * (fma(t_1, cos(x), (0.7639320225002103 * cos(y))) / 2.0)));
	} else if (x <= 0.039) {
		tmp = fma(((t_0 - (sin(y) * 0.0625)) * sqrt(2.0)), ((sin(y) - (t_0 * 0.0625)) * t_2), 2.0) * (1.0 / (fma(((3.0 - sqrt(5.0)) * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0)) * 3.0));
	} else {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / (3.0 * fma((((sqrt(5.0) - 3.0) * cos(y)) - (t_1 * cos(x))), -0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.07)
		tmp = Float64(fma(Float64(t_2 * sqrt(2.0)), Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))) / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(fma(Float64(Float64(t_0 - Float64(sin(y) * 0.0625)) * sqrt(2.0)), Float64(Float64(sin(y) - Float64(t_0 * 0.0625)) * t_2), 2.0) * Float64(1.0 / Float64(fma(Float64(Float64(3.0 - sqrt(5.0)) * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0)) * 3.0)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0))) / Float64(3.0 * fma(Float64(Float64(Float64(sqrt(5.0) - 3.0) * cos(y)) - Float64(t_1 * cos(x))), -0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.07], N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(N[(N[(t$95$0 - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(t$95$0 * 0.0625), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / N[(N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * 0.6180339887498949 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.070000000000000007

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-sin.f64 62.6%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -0.070000000000000007 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.1%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 51.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 51.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 51.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right) \cdot 3}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2}} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{\left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2}\right)\right)} \]

      11. div-add N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}}{2}\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, \frac{-1}{2}, 1\right)} \]

      2. lower-cos.f64 62.7%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.07:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{2 + \left(\left(t\_0 - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\left(\sin y - t\_0 \cdot 0.0625\right) \cdot t\_2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - t\_1 \cdot \cos x, -0.5, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y))))
   (if (<= x -0.07)
     (/
      (fma (* t_2 (sqrt 2.0)) (* -0.0625 (pow (sin x) 2.0)) 2.0)
      (+ 3.0 (* 3.0 (/ (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 2.0))))
     (if (<= x 0.039)
       (/
        (+
         2.0
         (*
          (* (- t_0 (* (sin y) 0.0625)) (sqrt 2.0))
          (* (- (sin y) (* t_0 0.0625)) t_2)))
        (*
         3.0
         (+
          (+ 1.0 (* 0.6180339887498949 (cos x)))
          (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
       (/
        (+
         2.0
         (*
          (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) 1.0)))
        (*
         3.0
         (fma
          (- (* (- (sqrt 5.0) 3.0) (cos y)) (* t_1 (cos x)))
          -0.5
          1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.07) {
		tmp = fma((t_2 * sqrt(2.0)), (-0.0625 * pow(sin(x), 2.0)), 2.0) / (3.0 + (3.0 * (fma(t_1, cos(x), (0.7639320225002103 * cos(y))) / 2.0)));
	} else if (x <= 0.039) {
		tmp = (2.0 + (((t_0 - (sin(y) * 0.0625)) * sqrt(2.0)) * ((sin(y) - (t_0 * 0.0625)) * t_2))) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	} else {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / (3.0 * fma((((sqrt(5.0) - 3.0) * cos(y)) - (t_1 * cos(x))), -0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.07)
		tmp = Float64(fma(Float64(t_2 * sqrt(2.0)), Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))) / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(t_0 - Float64(sin(y) * 0.0625)) * sqrt(2.0)) * Float64(Float64(sin(y) - Float64(t_0 * 0.0625)) * t_2))) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0))) / Float64(3.0 * fma(Float64(Float64(Float64(sqrt(5.0) - 3.0) * cos(y)) - Float64(t_1 * cos(x))), -0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.07], N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(2.0 + N[(N[(N[(t$95$0 - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(t$95$0 * 0.0625), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.070000000000000007

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-sin.f64 62.6%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -0.070000000000000007 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.1%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 51.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 51.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 51.3%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\left(\sin y - \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2}} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{\left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2}\right)\right)} \]

      11. div-add N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}}{2}\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, \frac{-1}{2}, 1\right)} \]

      2. lower-cos.f64 62.7%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.07:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_0 - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - t\_0 \cdot 0.0625\right) \cdot t\_2, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - t\_1 \cdot \cos x, -0.5, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y))))
   (if (<= x -0.07)
     (/
      (fma (* t_2 (sqrt 2.0)) (* -0.0625 (pow (sin x) 2.0)) 2.0)
      (+ 3.0 (* 3.0 (/ (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 2.0))))
     (if (<= x 0.039)
       (/
        (fma
         (* (- t_0 (* (sin y) 0.0625)) (sqrt 2.0))
         (* (- (sin y) (* t_0 0.0625)) t_2)
         2.0)
        (*
         (fma
          (* (- 3.0 (sqrt 5.0)) 0.5)
          (cos y)
          (fma (cos x) 0.6180339887498949 1.0))
         3.0))
       (/
        (+
         2.0
         (*
          (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) 1.0)))
        (*
         3.0
         (fma
          (- (* (- (sqrt 5.0) 3.0) (cos y)) (* t_1 (cos x)))
          -0.5
          1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.07) {
		tmp = fma((t_2 * sqrt(2.0)), (-0.0625 * pow(sin(x), 2.0)), 2.0) / (3.0 + (3.0 * (fma(t_1, cos(x), (0.7639320225002103 * cos(y))) / 2.0)));
	} else if (x <= 0.039) {
		tmp = fma(((t_0 - (sin(y) * 0.0625)) * sqrt(2.0)), ((sin(y) - (t_0 * 0.0625)) * t_2), 2.0) / (fma(((3.0 - sqrt(5.0)) * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0)) * 3.0);
	} else {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / (3.0 * fma((((sqrt(5.0) - 3.0) * cos(y)) - (t_1 * cos(x))), -0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.07)
		tmp = Float64(fma(Float64(t_2 * sqrt(2.0)), Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))) / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(fma(Float64(Float64(t_0 - Float64(sin(y) * 0.0625)) * sqrt(2.0)), Float64(Float64(sin(y) - Float64(t_0 * 0.0625)) * t_2), 2.0) / Float64(fma(Float64(Float64(3.0 - sqrt(5.0)) * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0)) * 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0))) / Float64(3.0 * fma(Float64(Float64(Float64(sqrt(5.0) - 3.0) * cos(y)) - Float64(t_1 * cos(x))), -0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.07], N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(N[(N[(t$95$0 - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(t$95$0 * 0.0625), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * 0.6180339887498949 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.070000000000000007

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-sin.f64 62.6%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -0.070000000000000007 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.1%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 51.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 51.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 51.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right) \cdot 3}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2}} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{\left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2}\right)\right)} \]

      11. div-add N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}}{2}\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, \frac{-1}{2}, 1\right)} \]

      2. lower-cos.f64 62.7%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.07:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_0 - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - t\_0 \cdot 0.0625\right) \cdot t\_2, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - t\_1 \cdot \cos x, -0.5, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y))))
   (if (<= x -0.07)
     (/
      (fma (* t_2 (sqrt 2.0)) (* -0.0625 (pow (sin x) 2.0)) 2.0)
      (+ 3.0 (* 3.0 (/ (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 2.0))))
     (if (<= x 0.039)
       (/
        (*
         (fma
          (* (- t_0 (* (sin y) 0.0625)) (sqrt 2.0))
          (* (- (sin y) (* t_0 0.0625)) t_2)
          2.0)
         0.3333333333333333)
        (fma
         (* (- 3.0 (sqrt 5.0)) 0.5)
         (cos y)
         (fma (cos x) 0.6180339887498949 1.0)))
       (/
        (+
         2.0
         (*
          (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) 1.0)))
        (*
         3.0
         (fma
          (- (* (- (sqrt 5.0) 3.0) (cos y)) (* t_1 (cos x)))
          -0.5
          1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.07) {
		tmp = fma((t_2 * sqrt(2.0)), (-0.0625 * pow(sin(x), 2.0)), 2.0) / (3.0 + (3.0 * (fma(t_1, cos(x), (0.7639320225002103 * cos(y))) / 2.0)));
	} else if (x <= 0.039) {
		tmp = (fma(((t_0 - (sin(y) * 0.0625)) * sqrt(2.0)), ((sin(y) - (t_0 * 0.0625)) * t_2), 2.0) * 0.3333333333333333) / fma(((3.0 - sqrt(5.0)) * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0));
	} else {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / (3.0 * fma((((sqrt(5.0) - 3.0) * cos(y)) - (t_1 * cos(x))), -0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.07)
		tmp = Float64(fma(Float64(t_2 * sqrt(2.0)), Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))) / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(Float64(fma(Float64(Float64(t_0 - Float64(sin(y) * 0.0625)) * sqrt(2.0)), Float64(Float64(sin(y) - Float64(t_0 * 0.0625)) * t_2), 2.0) * 0.3333333333333333) / fma(Float64(Float64(3.0 - sqrt(5.0)) * 0.5), cos(y), fma(cos(x), 0.6180339887498949, 1.0)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0))) / Float64(3.0 * fma(Float64(Float64(Float64(sqrt(5.0) - 3.0) * cos(y)) - Float64(t_1 * cos(x))), -0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.07], N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(N[(N[(N[(t$95$0 - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(t$95$0 * 0.0625), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * 0.6180339887498949 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.070000000000000007

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-sin.f64 62.6%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -0.070000000000000007 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.1%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 51.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-pow.f64 51.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 51.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 51.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}, \left(\sin y - \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right)}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2}} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{\left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2}\right)\right)} \]

      11. div-add N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}}{2}\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, \frac{-1}{2}, 1\right)} \]

      2. lower-cos.f64 62.7%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.07:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_0, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - t\_0 \cdot \cos x, -0.5, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- (cos x) (cos y))))
   (if (<= x -0.07)
     (/
      (fma (* t_1 (sqrt 2.0)) (* -0.0625 (pow (sin x) 2.0)) 2.0)
      (+ 3.0 (* 3.0 (/ (fma t_0 (cos x) (* 0.7639320225002103 (cos y))) 2.0))))
     (if (<= x 0.039)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- x (/ (sin y) 16.0))) (- (sin y) (/ x 16.0)))
          t_1))
        (*
         3.0
         (+
          (+ 1.0 (* (/ t_0 2.0) (cos x)))
          (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
       (/
        (+
         2.0
         (*
          (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) 1.0)))
        (*
         3.0
         (fma
          (- (* (- (sqrt 5.0) 3.0) (cos y)) (* t_0 (cos x)))
          -0.5
          1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.07) {
		tmp = fma((t_1 * sqrt(2.0)), (-0.0625 * pow(sin(x), 2.0)), 2.0) / (3.0 + (3.0 * (fma(t_0, cos(x), (0.7639320225002103 * cos(y))) / 2.0)));
	} else if (x <= 0.039) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))) * t_1)) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	} else {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / (3.0 * fma((((sqrt(5.0) - 3.0) * cos(y)) - (t_0 * cos(x))), -0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.07)
		tmp = Float64(fma(Float64(t_1 * sqrt(2.0)), Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_0, cos(x), Float64(0.7639320225002103 * cos(y))) / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x / 16.0))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0))) / Float64(3.0 * fma(Float64(Float64(Float64(sqrt(5.0) - 3.0) * cos(y)) - Float64(t_0 * cos(x))), -0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.07], N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.070000000000000007

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-sin.f64 62.6%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -0.070000000000000007 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2}} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{\left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2}\right)\right)} \]

      11. div-add N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}}{2}\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, \frac{-1}{2}, 1\right)} \]

      2. lower-cos.f64 62.7%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 0.7639320225002103 \cdot \cos y\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_0, \cos x, t\_1\right)}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{-2 - \left(1 - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)\right)}{\mathsf{fma}\left(-3, \mathsf{fma}\left(1, t\_0, t\_1\right) \cdot 0.5, -3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - t\_0 \cdot \cos x, -0.5, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (* 0.7639320225002103 (cos y))))
   (if (<= x -7.2e+15)
     (/
      (fma
       (* (- (cos x) (cos y)) (sqrt 2.0))
       (* -0.0625 (pow (sin x) 2.0))
       2.0)
      (+ 3.0 (* 3.0 (/ (fma t_0 (cos x) t_1) 2.0))))
     (if (<= x 0.039)
       (/
        (-
         -2.0
         (*
          (- 1.0 (cos y))
          (*
           (fma (sin x) -0.0625 (sin y))
           (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))))
        (fma -3.0 (* (fma 1.0 t_0 t_1) 0.5) -3.0))
       (/
        (+
         2.0
         (*
          (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) 1.0)))
        (*
         3.0
         (fma
          (- (* (- (sqrt 5.0) 3.0) (cos y)) (* t_0 (cos x)))
          -0.5
          1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 0.7639320225002103 * cos(y);
	double tmp;
	if (x <= -7.2e+15) {
		tmp = fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * pow(sin(x), 2.0)), 2.0) / (3.0 + (3.0 * (fma(t_0, cos(x), t_1) / 2.0)));
	} else if (x <= 0.039) {
		tmp = (-2.0 - ((1.0 - cos(y)) * (fma(sin(x), -0.0625, sin(y)) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))))) / fma(-3.0, (fma(1.0, t_0, t_1) * 0.5), -3.0);
	} else {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / (3.0 * fma((((sqrt(5.0) - 3.0) * cos(y)) - (t_0 * cos(x))), -0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(0.7639320225002103 * cos(y))
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_0, cos(x), t_1) / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(Float64(-2.0 - Float64(Float64(1.0 - cos(y)) * Float64(fma(sin(x), -0.0625, sin(y)) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))))) / fma(-3.0, Float64(fma(1.0, t_0, t_1) * 0.5), -3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0))) / Float64(3.0 * fma(Float64(Float64(Float64(sqrt(5.0) - 3.0) * cos(y)) - Float64(t_0 * cos(x))), -0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.2e+15], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(-2.0 - N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-3.0 * N[(N[(1.0 * t$95$0 + t$95$1), $MachinePrecision] * 0.5), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e15

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-sin.f64 62.6%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -7.2e15 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   12. Applied rewrites 60.2%

       \[\leadsto \color{blue}{\frac{-2 - \left(1 - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)\right)}{\mathsf{fma}\left(-3, \mathsf{fma}\left(1, \sqrt{5} - 1, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, -3\right)}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 64.3%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2}} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \color{blue}{\frac{\left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2} + \frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2}\right)\right)} \]

      11. div-add N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right) \cdot \cos y}{2}}\right)} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}}{2}\right)} \]

   6. Applied rewrites 64.3%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, \frac{-1}{2}, 1\right)} \]

      2. lower-cos.f64 62.7%

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]

   9. Applied rewrites 62.7%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y - \left(\sqrt{5} - 1\right) \cdot \cos x, -0.5, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.7639320225002103 \cdot \cos y\\ t_3 := \mathsf{fma}\left(t\_1, \cos x, t\_2\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot t\_0, 2\right)}{3 + 3 \cdot \frac{t\_3}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{-2 - \left(1 - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)\right)}{\mathsf{fma}\left(-3, \mathsf{fma}\left(1, t\_1, t\_2\right) \cdot 0.5, -3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, t\_3 \cdot 0.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* 0.7639320225002103 (cos y)))
        (t_3 (fma t_1 (cos x) t_2)))
   (if (<= x -7.2e+15)
     (/
      (fma (* (- (cos x) (cos y)) (sqrt 2.0)) (* -0.0625 t_0) 2.0)
      (+ 3.0 (* 3.0 (/ t_3 2.0))))
     (if (<= x 0.039)
       (/
        (-
         -2.0
         (*
          (- 1.0 (cos y))
          (*
           (fma (sin x) -0.0625 (sin y))
           (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))))
        (fma -3.0 (* (fma 1.0 t_1 t_2) 0.5) -3.0))
       (/
        (+ 2.0 (* -0.0625 (* t_0 (* (sqrt 2.0) (- (cos x) 1.0)))))
        (fma 3.0 (* t_3 0.5) 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.7639320225002103 * cos(y);
	double t_3 = fma(t_1, cos(x), t_2);
	double tmp;
	if (x <= -7.2e+15) {
		tmp = fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * t_0), 2.0) / (3.0 + (3.0 * (t_3 / 2.0)));
	} else if (x <= 0.039) {
		tmp = (-2.0 - ((1.0 - cos(y)) * (fma(sin(x), -0.0625, sin(y)) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))))) / fma(-3.0, (fma(1.0, t_1, t_2) * 0.5), -3.0);
	} else {
		tmp = (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(3.0, (t_3 * 0.5), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.7639320225002103 * cos(y))
	t_3 = fma(t_1, cos(x), t_2)
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * t_0), 2.0) / Float64(3.0 + Float64(3.0 * Float64(t_3 / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(Float64(-2.0 - Float64(Float64(1.0 - cos(y)) * Float64(fma(sin(x), -0.0625, sin(y)) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))))) / fma(-3.0, Float64(fma(1.0, t_1, t_2) * 0.5), -3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(3.0, Float64(t_3 * 0.5), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[x, -7.2e+15], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(-2.0 - N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-3.0 * N[(N[(1.0 * t$95$1 + t$95$2), $MachinePrecision] * 0.5), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e15

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-sin.f64 62.6%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -7.2e15 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   12. Applied rewrites 60.2%

       \[\leadsto \color{blue}{\frac{-2 - \left(1 - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)\right)}{\mathsf{fma}\left(-3, \mathsf{fma}\left(1, \sqrt{5} - 1, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, -3\right)}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      9. lower-cos.f64 62.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 0.7639320225002103 \cdot \cos y\\ t_2 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot t\_0, 2\right)}{3 + 3 \cdot \frac{t\_2}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + t\_1\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, t\_2 \cdot 0.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (* 0.7639320225002103 (cos y)))
        (t_2 (fma (- (sqrt 5.0) 1.0) (cos x) t_1)))
   (if (<= x -7.2e+15)
     (/
      (fma (* (- (cos x) (cos y)) (sqrt 2.0)) (* -0.0625 t_0) 2.0)
      (+ 3.0 (* 3.0 (/ t_2 2.0))))
     (if (<= x 0.039)
       (/
        (fma
         (* (- 1.0 (cos y)) (sqrt 2.0))
         (* (fma (sin x) -0.0625 (sin y)) (fma (sin y) -0.0625 (sin x)))
         2.0)
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) t_1) 1.0))))
       (/
        (+ 2.0 (* -0.0625 (* t_0 (* (sqrt 2.0) (- (cos x) 1.0)))))
        (fma 3.0 (* t_2 0.5) 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = 0.7639320225002103 * cos(y);
	double t_2 = fma((sqrt(5.0) - 1.0), cos(x), t_1);
	double tmp;
	if (x <= -7.2e+15) {
		tmp = fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * t_0), 2.0) / (3.0 + (3.0 * (t_2 / 2.0)));
	} else if (x <= 0.039) {
		tmp = fma(((1.0 - cos(y)) * sqrt(2.0)), (fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / (3.0 + (1.5 * ((sqrt(5.0) + t_1) - 1.0)));
	} else {
		tmp = (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(3.0, (t_2 * 0.5), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(0.7639320225002103 * cos(y))
	t_2 = fma(Float64(sqrt(5.0) - 1.0), cos(x), t_1)
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * t_0), 2.0) / Float64(3.0 + Float64(3.0 * Float64(t_2 / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + t_1) - 1.0))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(3.0, Float64(t_2 * 0.5), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[x, -7.2e+15], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + t$95$1), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e15

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-sin.f64 62.6%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -7.2e15 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + \color{blue}{\frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \color{blue}{\left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - \color{blue}{1}\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)} \]

       6. lower-cos.f64 60.1%

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)} \]

   13. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + \color{blue}{1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      9. lower-cos.f64 62.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.7639320225002103 \cdot \cos y\\ t_3 := \mathsf{fma}\left(t\_1, \cos x, t\_2\right)\\ \mathbf{if}\;x \leq -215:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot t\_0, 2\right)}{3 + 3 \cdot \frac{t\_3}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, 1, t\_2\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, t\_3 \cdot 0.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* 0.7639320225002103 (cos y)))
        (t_3 (fma t_1 (cos x) t_2)))
   (if (<= x -215.0)
     (/
      (fma (* (- (cos x) (cos y)) (sqrt 2.0)) (* -0.0625 t_0) 2.0)
      (+ 3.0 (* 3.0 (/ t_3 2.0))))
     (if (<= x 0.039)
       (/
        (fma
         (* (- 1.0 (cos y)) (sqrt 2.0))
         (fma
          -0.0625
          (pow (sin y) 2.0)
          (* x (+ (sin y) (* 0.00390625 (sin y)))))
         2.0)
        (+ 3.0 (* 3.0 (/ (fma t_1 1.0 t_2) 2.0))))
       (/
        (+ 2.0 (* -0.0625 (* t_0 (* (sqrt 2.0) (- (cos x) 1.0)))))
        (fma 3.0 (* t_3 0.5) 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.7639320225002103 * cos(y);
	double t_3 = fma(t_1, cos(x), t_2);
	double tmp;
	if (x <= -215.0) {
		tmp = fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * t_0), 2.0) / (3.0 + (3.0 * (t_3 / 2.0)));
	} else if (x <= 0.039) {
		tmp = fma(((1.0 - cos(y)) * sqrt(2.0)), fma(-0.0625, pow(sin(y), 2.0), (x * (sin(y) + (0.00390625 * sin(y))))), 2.0) / (3.0 + (3.0 * (fma(t_1, 1.0, t_2) / 2.0)));
	} else {
		tmp = (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(3.0, (t_3 * 0.5), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.7639320225002103 * cos(y))
	t_3 = fma(t_1, cos(x), t_2)
	tmp = 0.0
	if (x <= -215.0)
		tmp = Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * t_0), 2.0) / Float64(3.0 + Float64(3.0 * Float64(t_3 / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), fma(-0.0625, (sin(y) ^ 2.0), Float64(x * Float64(sin(y) + Float64(0.00390625 * sin(y))))), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, 1.0, t_2) / 2.0))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(3.0, Float64(t_3 * 0.5), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[x, -215.0], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] + N[(0.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * 1.0 + t$95$2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -215

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-sin.f64 62.6%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -215 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \color{blue}{{\sin y}^{2}}, x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       2. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, {\sin y}^{\color{blue}{2}}, x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       3. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, {\sin y}^{2}, x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, {\sin y}^{2}, x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       5. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, {\sin y}^{2}, x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, {\sin y}^{2}, x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, {\sin y}^{2}, x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       8. lower-sin.f64 55.7%

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   13. Applied rewrites 55.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      9. lower-cos.f64 62.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.7639320225002103 \cdot \cos y\\ t_3 := \mathsf{fma}\left(t\_1, \cos x, t\_2\right)\\ \mathbf{if}\;x \leq -215:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot t\_0, 2\right)}{3 + 3 \cdot \frac{t\_3}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(x + -0.0625 \cdot \sin y\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, 1, t\_2\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, t\_3 \cdot 0.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* 0.7639320225002103 (cos y)))
        (t_3 (fma t_1 (cos x) t_2)))
   (if (<= x -215.0)
     (/
      (fma (* (- (cos x) (cos y)) (sqrt 2.0)) (* -0.0625 t_0) 2.0)
      (+ 3.0 (* 3.0 (/ t_3 2.0))))
     (if (<= x 0.039)
       (/
        (fma
         (* (- 1.0 (cos y)) (sqrt 2.0))
         (* (fma (sin x) -0.0625 (sin y)) (+ x (* -0.0625 (sin y))))
         2.0)
        (+ 3.0 (* 3.0 (/ (fma t_1 1.0 t_2) 2.0))))
       (/
        (+ 2.0 (* -0.0625 (* t_0 (* (sqrt 2.0) (- (cos x) 1.0)))))
        (fma 3.0 (* t_3 0.5) 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.7639320225002103 * cos(y);
	double t_3 = fma(t_1, cos(x), t_2);
	double tmp;
	if (x <= -215.0) {
		tmp = fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * t_0), 2.0) / (3.0 + (3.0 * (t_3 / 2.0)));
	} else if (x <= 0.039) {
		tmp = fma(((1.0 - cos(y)) * sqrt(2.0)), (fma(sin(x), -0.0625, sin(y)) * (x + (-0.0625 * sin(y)))), 2.0) / (3.0 + (3.0 * (fma(t_1, 1.0, t_2) / 2.0)));
	} else {
		tmp = (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(3.0, (t_3 * 0.5), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.7639320225002103 * cos(y))
	t_3 = fma(t_1, cos(x), t_2)
	tmp = 0.0
	if (x <= -215.0)
		tmp = Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * t_0), 2.0) / Float64(3.0 + Float64(3.0 * Float64(t_3 / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(x + Float64(-0.0625 * sin(y)))), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, 1.0, t_2) / 2.0))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(3.0, Float64(t_3 * 0.5), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[x, -215.0], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * 1.0 + t$95$2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -215

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-sin.f64 62.6%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -215 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\left(x + \frac{-1}{16} \cdot \sin y\right)}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(x + \color{blue}{\frac{-1}{16} \cdot \sin y}\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(x + \frac{-1}{16} \cdot \color{blue}{\sin y}\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       3. lower-sin.f64 55.7%

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(x + -0.0625 \cdot \sin y\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   13. Applied rewrites 55.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\left(x + -0.0625 \cdot \sin y\right)}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      9. lower-cos.f64 62.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 0.7639320225002103 \cdot \cos y\\ t_2 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\\ t_3 := \mathsf{fma}\left(t\_0, \cos x, t\_1\right)\\ \mathbf{if}\;x \leq -215:\\ \;\;\;\;\frac{t\_2}{3 + 3 \cdot \frac{t\_3}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(x + -0.0625 \cdot \sin y\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_0, 1, t\_1\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(3, t\_3 \cdot 0.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* 0.7639320225002103 (cos y)))
        (t_2
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
        (t_3 (fma t_0 (cos x) t_1)))
   (if (<= x -215.0)
     (/ t_2 (+ 3.0 (* 3.0 (/ t_3 2.0))))
     (if (<= x 0.039)
       (/
        (fma
         (* (- 1.0 (cos y)) (sqrt 2.0))
         (* (fma (sin x) -0.0625 (sin y)) (+ x (* -0.0625 (sin y))))
         2.0)
        (+ 3.0 (* 3.0 (/ (fma t_0 1.0 t_1) 2.0))))
       (/ t_2 (fma 3.0 (* t_3 0.5) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 0.7639320225002103 * cos(y);
	double t_2 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))));
	double t_3 = fma(t_0, cos(x), t_1);
	double tmp;
	if (x <= -215.0) {
		tmp = t_2 / (3.0 + (3.0 * (t_3 / 2.0)));
	} else if (x <= 0.039) {
		tmp = fma(((1.0 - cos(y)) * sqrt(2.0)), (fma(sin(x), -0.0625, sin(y)) * (x + (-0.0625 * sin(y)))), 2.0) / (3.0 + (3.0 * (fma(t_0, 1.0, t_1) / 2.0)));
	} else {
		tmp = t_2 / fma(3.0, (t_3 * 0.5), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(0.7639320225002103 * cos(y))
	t_2 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))
	t_3 = fma(t_0, cos(x), t_1)
	tmp = 0.0
	if (x <= -215.0)
		tmp = Float64(t_2 / Float64(3.0 + Float64(3.0 * Float64(t_3 / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(x + Float64(-0.0625 * sin(y)))), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_0, 1.0, t_1) / 2.0))));
	else
		tmp = Float64(t_2 / fma(3.0, Float64(t_3 * 0.5), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[x, -215.0], N[(t$95$2 / N[(3.0 + N[(3.0 * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 * N[(t$95$3 * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -215

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      9. lower-cos.f64 62.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -215 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\left(x + \frac{-1}{16} \cdot \sin y\right)}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(x + \color{blue}{\frac{-1}{16} \cdot \sin y}\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(x + \frac{-1}{16} \cdot \color{blue}{\sin y}\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       3. lower-sin.f64 55.7%

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(x + -0.0625 \cdot \sin y\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   13. Applied rewrites 55.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\left(x + -0.0625 \cdot \sin y\right)}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      9. lower-cos.f64 62.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 0.7639320225002103 \cdot \cos y\\ t_2 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\\ t_3 := \mathsf{fma}\left(t\_0, \cos x, t\_1\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_2}{3 + 3 \cdot \frac{t\_3}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(-0.0625 \cdot \sin y\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_0, 1, t\_1\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(3, t\_3 \cdot 0.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* 0.7639320225002103 (cos y)))
        (t_2
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
        (t_3 (fma t_0 (cos x) t_1)))
   (if (<= x -7.2e+15)
     (/ t_2 (+ 3.0 (* 3.0 (/ t_3 2.0))))
     (if (<= x 0.039)
       (/
        (fma
         (* (- 1.0 (cos y)) (sqrt 2.0))
         (* (fma (sin x) -0.0625 (sin y)) (* -0.0625 (sin y)))
         2.0)
        (+ 3.0 (* 3.0 (/ (fma t_0 1.0 t_1) 2.0))))
       (/ t_2 (fma 3.0 (* t_3 0.5) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 0.7639320225002103 * cos(y);
	double t_2 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))));
	double t_3 = fma(t_0, cos(x), t_1);
	double tmp;
	if (x <= -7.2e+15) {
		tmp = t_2 / (3.0 + (3.0 * (t_3 / 2.0)));
	} else if (x <= 0.039) {
		tmp = fma(((1.0 - cos(y)) * sqrt(2.0)), (fma(sin(x), -0.0625, sin(y)) * (-0.0625 * sin(y))), 2.0) / (3.0 + (3.0 * (fma(t_0, 1.0, t_1) / 2.0)));
	} else {
		tmp = t_2 / fma(3.0, (t_3 * 0.5), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(0.7639320225002103 * cos(y))
	t_2 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))
	t_3 = fma(t_0, cos(x), t_1)
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = Float64(t_2 / Float64(3.0 + Float64(3.0 * Float64(t_3 / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(-0.0625 * sin(y))), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_0, 1.0, t_1) / 2.0))));
	else
		tmp = Float64(t_2 / fma(3.0, Float64(t_3 * 0.5), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[x, -7.2e+15], N[(t$95$2 / N[(3.0 + N[(3.0 * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 * N[(t$95$3 * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e15

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      9. lower-cos.f64 62.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -7.2e15 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \sin y\right)}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\frac{-1}{16} \cdot \color{blue}{\sin y}\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       2. lower-sin.f64 59.7%

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(-0.0625 \cdot \sin y\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   13. Applied rewrites 59.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\left(-0.0625 \cdot \sin y\right)}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      9. lower-cos.f64 62.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 79.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 0.7639320225002103 \cdot \cos y\\ t_2 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\\ t_3 := \mathsf{fma}\left(t\_0, \cos x, t\_1\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_2}{3 + 3 \cdot \frac{t\_3}{2}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_0, 1, t\_1\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(3, t\_3 \cdot 0.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* 0.7639320225002103 (cos y)))
        (t_2
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
        (t_3 (fma t_0 (cos x) t_1)))
   (if (<= x -7.2e+15)
     (/ t_2 (+ 3.0 (* 3.0 (/ t_3 2.0))))
     (if (<= x 0.039)
       (/
        (fma (* (- 1.0 (cos y)) (sqrt 2.0)) (* -0.0625 (pow (sin y) 2.0)) 2.0)
        (+ 3.0 (* 3.0 (/ (fma t_0 1.0 t_1) 2.0))))
       (/ t_2 (fma 3.0 (* t_3 0.5) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 0.7639320225002103 * cos(y);
	double t_2 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))));
	double t_3 = fma(t_0, cos(x), t_1);
	double tmp;
	if (x <= -7.2e+15) {
		tmp = t_2 / (3.0 + (3.0 * (t_3 / 2.0)));
	} else if (x <= 0.039) {
		tmp = fma(((1.0 - cos(y)) * sqrt(2.0)), (-0.0625 * pow(sin(y), 2.0)), 2.0) / (3.0 + (3.0 * (fma(t_0, 1.0, t_1) / 2.0)));
	} else {
		tmp = t_2 / fma(3.0, (t_3 * 0.5), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(0.7639320225002103 * cos(y))
	t_2 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))
	t_3 = fma(t_0, cos(x), t_1)
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = Float64(t_2 / Float64(3.0 + Float64(3.0 * Float64(t_3 / 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_0, 1.0, t_1) / 2.0))));
	else
		tmp = Float64(t_2 / fma(3.0, Float64(t_3 * 0.5), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[x, -7.2e+15], N[(t$95$2 / N[(3.0 + N[(3.0 * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 * N[(t$95$3 * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e15

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      9. lower-cos.f64 62.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if -7.2e15 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       2. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin y}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       3. lower-sin.f64 59.7%

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   13. Applied rewrites 59.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin y}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      9. lower-cos.f64 62.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 79.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 0.7639320225002103 \cdot \cos y\\ t_2 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_0, \cos x, t\_1\right) \cdot 0.5, 3\right)}\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_0, 1, t\_1\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* 0.7639320225002103 (cos y)))
        (t_2
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
          (fma 3.0 (* (fma t_0 (cos x) t_1) 0.5) 3.0))))
   (if (<= x -7.2e+15)
     t_2
     (if (<= x 0.039)
       (/
        (fma (* (- 1.0 (cos y)) (sqrt 2.0)) (* -0.0625 (pow (sin y) 2.0)) 2.0)
        (+ 3.0 (* 3.0 (/ (fma t_0 1.0 t_1) 2.0))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 0.7639320225002103 * cos(y);
	double t_2 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(3.0, (fma(t_0, cos(x), t_1) * 0.5), 3.0);
	double tmp;
	if (x <= -7.2e+15) {
		tmp = t_2;
	} else if (x <= 0.039) {
		tmp = fma(((1.0 - cos(y)) * sqrt(2.0)), (-0.0625 * pow(sin(y), 2.0)), 2.0) / (3.0 + (3.0 * (fma(t_0, 1.0, t_1) / 2.0)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(0.7639320225002103 * cos(y))
	t_2 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(3.0, Float64(fma(t_0, cos(x), t_1) * 0.5), 3.0))
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = t_2;
	elseif (x <= 0.039)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_0, 1.0, t_1) / 2.0))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.2e+15], t$95$2, If[LessEqual[x, 0.039], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2e15 or 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right) + 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right)} + 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      5. lower-*.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \frac{-1}{16} + \sin y\right)}, \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\frac{-1}{16} \cdot \sin x} + \sin y\right), \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

      8. lower-fma.f64 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin x, \sin y\right)}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3\right)} \]

      9. lower-cos.f64 62.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   9. Applied rewrites 62.6%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3\right)} \]

   if -7.2e15 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       2. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin y}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       3. lower-sin.f64 59.7%

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   13. Applied rewrites 59.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin y}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_2, t\_0\right), 1\right)}{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot t\_1, 2\right)}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_2, 1, 0.7639320225002103 \cdot \cos y\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 1\right) + 1}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, 0.5 \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -7.2e+15)
     (/
      1.0
      (/
       (fma 0.5 (fma (cos x) t_2 t_0) 1.0)
       (*
        0.3333333333333333
        (fma -0.0625 (* (- 0.5 (* (cos (+ x x)) 0.5)) t_1) 2.0))))
     (if (<= x 0.039)
       (/
        (fma (* (- 1.0 (cos y)) (sqrt 2.0)) (* -0.0625 (pow (sin y) 2.0)) 2.0)
        (+ 3.0 (* 3.0 (/ (fma t_2 1.0 (* 0.7639320225002103 (cos y))) 2.0))))
       (*
        0.3333333333333333
        (/
         (+ (fma (* -0.0625 t_1) (- 0.5 (* 0.5 (cos (* 2.0 x)))) 1.0) 1.0)
         (+ 1.0 (fma 0.5 (* (cos x) t_2) (* 0.5 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -7.2e+15) {
		tmp = 1.0 / (fma(0.5, fma(cos(x), t_2, t_0), 1.0) / (0.3333333333333333 * fma(-0.0625, ((0.5 - (cos((x + x)) * 0.5)) * t_1), 2.0)));
	} else if (x <= 0.039) {
		tmp = fma(((1.0 - cos(y)) * sqrt(2.0)), (-0.0625 * pow(sin(y), 2.0)), 2.0) / (3.0 + (3.0 * (fma(t_2, 1.0, (0.7639320225002103 * cos(y))) / 2.0)));
	} else {
		tmp = 0.3333333333333333 * ((fma((-0.0625 * t_1), (0.5 - (0.5 * cos((2.0 * x)))), 1.0) + 1.0) / (1.0 + fma(0.5, (cos(x) * t_2), (0.5 * t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = Float64(1.0 / Float64(fma(0.5, fma(cos(x), t_2, t_0), 1.0) / Float64(0.3333333333333333 * fma(-0.0625, Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * t_1), 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_2, 1.0, Float64(0.7639320225002103 * cos(y))) / 2.0))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(fma(Float64(-0.0625 * t_1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 1.0) + 1.0) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), Float64(0.5 * t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -7.2e+15], N[(1.0 / N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(0.3333333333333333 * N[(-0.0625 * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$2 * 1.0 + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e15

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos \left(\mathsf{neg}\left(x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 43.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 43.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{\color{blue}{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 41.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. Applied rewrites 41.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 41.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. Applied rewrites 41.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   12. Applied rewrites 60.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)}{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}}} \]

   if -7.2e15 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       2. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin y}^{\color{blue}{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2}} \]

       3. lower-sin.f64 59.7%

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   13. Applied rewrites 59.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin y}^{2}}, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, 0.7639320225002103 \cdot \cos y\right)}{2}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \left(1 + 1\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 1\right) + 1}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 1\right) + 1}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 60.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 1\right) + 1}{\color{blue}{1} + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_2, t\_0\right), 1\right)}{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot t\_1, 2\right)}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 1\right) + 1}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, 0.5 \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -7.2e+15)
     (/
      1.0
      (/
       (fma 0.5 (fma (cos x) t_2 t_0) 1.0)
       (*
        0.3333333333333333
        (fma -0.0625 (* (- 0.5 (* (cos (+ x x)) 0.5)) t_1) 2.0))))
     (if (<= x 0.039)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* 0.7639320225002103 (cos y))) 1.0))))
       (*
        0.3333333333333333
        (/
         (+ (fma (* -0.0625 t_1) (- 0.5 (* 0.5 (cos (* 2.0 x)))) 1.0) 1.0)
         (+ 1.0 (fma 0.5 (* (cos x) t_2) (* 0.5 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -7.2e+15) {
		tmp = 1.0 / (fma(0.5, fma(cos(x), t_2, t_0), 1.0) / (0.3333333333333333 * fma(-0.0625, ((0.5 - (cos((x + x)) * 0.5)) * t_1), 2.0)));
	} else if (x <= 0.039) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + (0.7639320225002103 * cos(y))) - 1.0)));
	} else {
		tmp = 0.3333333333333333 * ((fma((-0.0625 * t_1), (0.5 - (0.5 * cos((2.0 * x)))), 1.0) + 1.0) / (1.0 + fma(0.5, (cos(x) * t_2), (0.5 * t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = Float64(1.0 / Float64(fma(0.5, fma(cos(x), t_2, t_0), 1.0) / Float64(0.3333333333333333 * fma(-0.0625, Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * t_1), 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(0.7639320225002103 * cos(y))) - 1.0))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(fma(Float64(-0.0625 * t_1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 1.0) + 1.0) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), Float64(0.5 * t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -7.2e+15], N[(1.0 / N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(0.3333333333333333 * N[(-0.0625 * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e15

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos \left(\mathsf{neg}\left(x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 43.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 43.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{\color{blue}{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 41.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. Applied rewrites 41.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 41.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. Applied rewrites 41.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   12. Applied rewrites 60.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)}{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}}} \]

   if -7.2e15 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]

   7. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \left(1 + 1\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 1\right) + 1}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 1\right) + 1}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 60.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 1\right) + 1}{\color{blue}{1} + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_2, t\_0\right), 1\right)}{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot t\_1, 2\right)}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, t\_0\right), 0.5, 1\right)}{\mathsf{fma}\left(-0.0625 \cdot t\_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -7.2e+15)
     (/
      1.0
      (/
       (fma 0.5 (fma (cos x) t_2 t_0) 1.0)
       (*
        0.3333333333333333
        (fma -0.0625 (* (- 0.5 (* (cos (+ x x)) 0.5)) t_1) 2.0))))
     (if (<= x 0.039)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* 0.7639320225002103 (cos y))) 1.0))))
       (*
        0.3333333333333333
        (/
         1.0
         (/
          (fma (fma t_2 (cos x) t_0) 0.5 1.0)
          (fma (* -0.0625 t_1) (- 0.5 (* 0.5 (cos (* 2.0 x)))) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -7.2e+15) {
		tmp = 1.0 / (fma(0.5, fma(cos(x), t_2, t_0), 1.0) / (0.3333333333333333 * fma(-0.0625, ((0.5 - (cos((x + x)) * 0.5)) * t_1), 2.0)));
	} else if (x <= 0.039) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + (0.7639320225002103 * cos(y))) - 1.0)));
	} else {
		tmp = 0.3333333333333333 * (1.0 / (fma(fma(t_2, cos(x), t_0), 0.5, 1.0) / fma((-0.0625 * t_1), (0.5 - (0.5 * cos((2.0 * x)))), 2.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = Float64(1.0 / Float64(fma(0.5, fma(cos(x), t_2, t_0), 1.0) / Float64(0.3333333333333333 * fma(-0.0625, Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * t_1), 2.0))));
	elseif (x <= 0.039)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(0.7639320225002103 * cos(y))) - 1.0))));
	else
		tmp = Float64(0.3333333333333333 * Float64(1.0 / Float64(fma(fma(t_2, cos(x), t_0), 0.5, 1.0) / fma(Float64(-0.0625 * t_1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -7.2e+15], N[(1.0 / N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(0.3333333333333333 * N[(-0.0625 * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(1.0 / N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] / N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e15

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos \left(\mathsf{neg}\left(x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 43.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 43.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{\color{blue}{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 41.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. Applied rewrites 41.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 41.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. Applied rewrites 41.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   12. Applied rewrites 60.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)}{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}}} \]

   if -7.2e15 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]

   7. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1}{\color{blue}{\frac{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1}{\color{blue}{\frac{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}} \]

      4. lower-unsound-/.f64 60.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{1}{\frac{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}} \]

   6. Applied rewrites 60.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)\\ t_1 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{0.3333333333333333 \cdot t\_0}}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_1} \cdot 0.3333333333333333\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          -0.0625
          (* (- 0.5 (* (cos (+ x x)) 0.5)) (* (- (cos x) 1.0) (sqrt 2.0)))
          2.0))
        (t_1
         (fma 0.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 1.0)))
   (if (<= x -7.2e+15)
     (/ 1.0 (/ t_1 (* 0.3333333333333333 t_0)))
     (if (<= x 0.039)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* 0.7639320225002103 (cos y))) 1.0))))
       (* (/ t_0 t_1) 0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, ((0.5 - (cos((x + x)) * 0.5)) * ((cos(x) - 1.0) * sqrt(2.0))), 2.0);
	double t_1 = fma(0.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 1.0);
	double tmp;
	if (x <= -7.2e+15) {
		tmp = 1.0 / (t_1 / (0.3333333333333333 * t_0));
	} else if (x <= 0.039) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + (0.7639320225002103 * cos(y))) - 1.0)));
	} else {
		tmp = (t_0 / t_1) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.0625, Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), 2.0)
	t_1 = fma(0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 1.0)
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = Float64(1.0 / Float64(t_1 / Float64(0.3333333333333333 * t_0)));
	elseif (x <= 0.039)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(0.7639320225002103 * cos(y))) - 1.0))));
	else
		tmp = Float64(Float64(t_0 / t_1) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.0625 * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -7.2e+15], N[(1.0 / N[(t$95$1 / N[(0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$1), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e15

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos \left(\mathsf{neg}\left(x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 43.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 43.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{\color{blue}{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 41.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. Applied rewrites 41.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 41.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. Applied rewrites 41.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   12. Applied rewrites 60.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)}{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}}} \]

   if -7.2e15 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]

   7. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos \left(\mathsf{neg}\left(x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 43.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 43.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{\color{blue}{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 41.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. Applied rewrites 41.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 41.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. Applied rewrites 41.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   12. Applied rewrites 60.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)\\ t_1 := \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot t\_1}{t\_0}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_0} \cdot 0.3333333333333333\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 1.0))
        (t_1
         (fma
          -0.0625
          (* (- 0.5 (* (cos (+ x x)) 0.5)) (* (- (cos x) 1.0) (sqrt 2.0)))
          2.0)))
   (if (<= x -7.2e+15)
     (/ (* 0.3333333333333333 t_1) t_0)
     (if (<= x 0.039)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* 0.7639320225002103 (cos y))) 1.0))))
       (* (/ t_1 t_0) 0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 1.0);
	double t_1 = fma(-0.0625, ((0.5 - (cos((x + x)) * 0.5)) * ((cos(x) - 1.0) * sqrt(2.0))), 2.0);
	double tmp;
	if (x <= -7.2e+15) {
		tmp = (0.3333333333333333 * t_1) / t_0;
	} else if (x <= 0.039) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + (0.7639320225002103 * cos(y))) - 1.0)));
	} else {
		tmp = (t_1 / t_0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 1.0)
	t_1 = fma(-0.0625, Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), 2.0)
	tmp = 0.0
	if (x <= -7.2e+15)
		tmp = Float64(Float64(0.3333333333333333 * t_1) / t_0);
	elseif (x <= 0.039)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(0.7639320225002103 * cos(y))) - 1.0))));
	else
		tmp = Float64(Float64(t_1 / t_0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[x, -7.2e+15], N[(N[(0.3333333333333333 * t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e15

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos \left(\mathsf{neg}\left(x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 43.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 43.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{\color{blue}{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 41.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. Applied rewrites 41.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 41.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. Applied rewrites 41.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   12. Applied rewrites 60.3%

       \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)}} \]

   if -7.2e15 < x < 0.0389999999999999999

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]

   7. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos \left(\mathsf{neg}\left(x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 43.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 43.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{\color{blue}{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-PI.f64 41.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. Applied rewrites 41.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-*.f64 41.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. Applied rewrites 41.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   12. Applied rewrites 60.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 30: 60.3% accurate, 2.2× speedup

Math

\[\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (*
   0.3333333333333333
   (fma
    -0.0625
    (* (- 0.5 (* (cos (+ x x)) 0.5)) (* (- (cos x) 1.0) (sqrt 2.0)))
    2.0))
  (fma 0.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 1.0)))

C

double code(double x, double y) {
	return (0.3333333333333333 * fma(-0.0625, ((0.5 - (cos((x + x)) * 0.5)) * ((cos(x) - 1.0) * sqrt(2.0))), 2.0)) / fma(0.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 1.0);
}

Julia

function code(x, y)
	return Float64(Float64(0.3333333333333333 * fma(-0.0625, Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), 2.0)) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 1.0))
end

Wolfram

code[x_, y_] := N[(N[(0.3333333333333333 * N[(-0.0625 * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 60.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos \left(\mathsf{neg}\left(x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. sin-+PI/2-rev N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. lower-neg.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   7. mult-flip N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. lower-PI.f64 43.7%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

6. Applied rewrites 43.7%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. sin-+PI/2-rev N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{\color{blue}{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. lower-neg.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   7. mult-flip N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. lower-PI.f64 41.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

8. Applied rewrites 41.3%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. lift-sin.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. lift-sin.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. sqr-sin-a N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. lower-*.f64 41.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

10. Applied rewrites 41.3%

    \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

12. Applied rewrites 60.3%

    \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)}} \]
13. Add Preprocessing

Alternative 31: 60.3% accurate, 2.2× speedup

Math

\[\frac{\mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    -0.0625
    (* (- 0.5 (* (cos (+ x x)) 0.5)) (* (- (cos x) 1.0) (sqrt 2.0)))
    2.0)
   (fma 0.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 1.0))
  0.3333333333333333))

C

double code(double x, double y) {
	return (fma(-0.0625, ((0.5 - (cos((x + x)) * 0.5)) * ((cos(x) - 1.0) * sqrt(2.0))), 2.0) / fma(0.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(fma(-0.0625, Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), 2.0) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(N[(-0.0625 * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 60.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos \left(\mathsf{neg}\left(x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. sin-+PI/2-rev N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. lower-neg.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   7. mult-flip N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. lower-PI.f64 43.7%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

6. Applied rewrites 43.7%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. sin-+PI/2-rev N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{\color{blue}{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. lower-neg.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   7. mult-flip N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. lower-PI.f64 41.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

8. Applied rewrites 41.3%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. lift-sin.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. lift-sin.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. sqr-sin-a N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\left(-x\right) + \pi \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. lower-*.f64 41.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

10. Applied rewrites 41.3%

    \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin \left(\left(-x\right) + \pi \cdot 0.5\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\left(-x\right) + \pi \cdot 0.5\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

12. Applied rewrites 60.3%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
13. Add Preprocessing

Alternative 32: 43.3% accurate, 5.1× speedup

Math

\[0.3333333333333333 \cdot \frac{2}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   2.0
   (+
    1.0
    (fma 0.5 (* (cos x) (- (sqrt 5.0) 1.0)) (* 0.5 (- 3.0 (sqrt 5.0))))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * (2.0 / (1.0 + fma(0.5, (cos(x) * (sqrt(5.0) - 1.0)), (0.5 * (3.0 - sqrt(5.0))))));
}

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(2.0 / Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(sqrt(5.0) - 1.0)), Float64(0.5 * Float64(3.0 - sqrt(5.0)))))))
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(2.0 / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 60.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto 0.3333333333333333 \cdot \frac{2}{\color{blue}{1} + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 43.3%

   \[\leadsto 0.3333333333333333 \cdot \frac{2}{\color{blue}{1} + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
8. Add Preprocessing

Alternative 33: 40.8% accurate, 316.7× speedup

Math

\[0.3333333333333333 \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 60.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{2}{3}}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \color{blue}{\sqrt{5}}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. lower-sqrt.f64 40.8%

      \[\leadsto \frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, 3 - \sqrt{5}, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)} \]

7. Applied rewrites 40.8%

   \[\leadsto \frac{0.6666666666666666}{\color{blue}{1 + \mathsf{fma}\left(0.5, 3 - \sqrt{5}, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]

8. Evaluated real constant 40.8%

   \[\leadsto 0.3333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (- (sin y) (/ (sin x) 16.0))) (- (cos x) (cos y)))) (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))