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

Percentage Accurate: 99.3% → 99.3%

Time: 12.3s

Alternatives: 31

Speedup: 1.0×

Specification

Math

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

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

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

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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \mathsf{fma}\left(\sin y \cdot -0.0625, \sqrt{2}, \sin x \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (- (cos y) (cos x))
   (*
    (fma -0.0625 (sin x) (sin y))
    (fma (* (sin y) -0.0625) (sqrt 2.0) (* (sin x) (sqrt 2.0))))
   -2.0)
  (-
   (fma
    (fma (- (sqrt 5.0) 3.0) (cos y) (* (- 1.0 (sqrt 5.0)) (cos x)))
    -1.5
    3.0))))

C

double code(double x, double y) {
	return fma((cos(y) - cos(x)), (fma(-0.0625, sin(x), sin(y)) * fma((sin(y) * -0.0625), sqrt(2.0), (sin(x) * sqrt(2.0)))), -2.0) / -fma(fma((sqrt(5.0) - 3.0), cos(y), ((1.0 - sqrt(5.0)) * cos(x))), -1.5, 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(cos(y) - cos(x)), Float64(fma(-0.0625, sin(x), sin(y)) * fma(Float64(sin(y) * -0.0625), sqrt(2.0), Float64(sin(x) * sqrt(2.0)))), -2.0) / Float64(-fma(fma(Float64(sqrt(5.0) - 3.0), cos(y), Float64(Float64(1.0 - sqrt(5.0)) * cos(x))), -1.5, 3.0)))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 99.3

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

9. Applied rewrites 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (- (cos y) (cos x))
   (*
    (fma (sin y) -0.0625 (sin x))
    (* (sqrt 2.0) (fma (sin x) -0.0625 (sin y))))
   -2.0)
  (-
   (fma
    (fma (- (sqrt 5.0) 3.0) (cos y) (* (- 1.0 (sqrt 5.0)) (cos x)))
    -1.5
    3.0))))

C

double code(double x, double y) {
	return fma((cos(y) - cos(x)), (fma(sin(y), -0.0625, sin(x)) * (sqrt(2.0) * fma(sin(x), -0.0625, sin(y)))), -2.0) / -fma(fma((sqrt(5.0) - 3.0), cos(y), ((1.0 - sqrt(5.0)) * cos(x))), -1.5, 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(cos(y) - cos(x)), Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sqrt(2.0) * fma(sin(x), -0.0625, sin(y)))), -2.0) / Float64(-fma(fma(Float64(sqrt(5.0) - 3.0), cos(y), Float64(Float64(1.0 - sqrt(5.0)) * cos(x))), -1.5, 3.0)))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-fma.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 99.3

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

   10. lift-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.3

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

9. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (- (cos y) (cos x))
   (*
    (fma -0.0625 (sin x) (sin y))
    (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)))
   -2.0)
  (-
   (fma
    (fma (- (sqrt 5.0) 3.0) (cos y) (* (- 1.0 (sqrt 5.0)) (cos x)))
    -1.5
    3.0))))

C

double code(double x, double y) {
	return fma((cos(y) - cos(x)), (fma(-0.0625, sin(x), sin(y)) * (fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / -fma(fma((sqrt(5.0) - 3.0), cos(y), ((1.0 - sqrt(5.0)) * cos(x))), -1.5, 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(cos(y) - cos(x)), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / Float64(-fma(fma(Float64(sqrt(5.0) - 3.0), cos(y), Float64(Float64(1.0 - sqrt(5.0)) * cos(x))), -1.5, 3.0)))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos y - \cos x\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2}, -2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (*
    (* (fma (sin x) -0.0625 (sin y)) (- (cos y) (cos x)))
    (fma (sin y) -0.0625 (sin x)))
   (sqrt 2.0)
   -2.0)
  (fma
   1.5
   (fma (- 1.0 (sqrt 5.0)) (cos x) (* (- (sqrt 5.0) 3.0) (cos y)))
   -3.0)))

C

double code(double x, double y) {
	return fma(((fma(sin(x), -0.0625, sin(y)) * (cos(y) - cos(x))) * fma(sin(y), -0.0625, sin(x))), sqrt(2.0), -2.0) / fma(1.5, fma((1.0 - sqrt(5.0)), cos(x), ((sqrt(5.0) - 3.0) * cos(y))), -3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(fma(sin(x), -0.0625, sin(y)) * Float64(cos(y) - cos(x))) * fma(sin(y), -0.0625, sin(x))), sqrt(2.0), -2.0) / fma(1.5, fma(Float64(1.0 - sqrt(5.0)), cos(x), Float64(Float64(sqrt(5.0) - 3.0) * cos(y))), -3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + -2.0), $MachinePrecision] / N[(1.5 * N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

9. Applied rewrites 99.3%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (* (fma -0.0625 (sin x) (sin y)) (- (cos x) (cos y))) (sqrt 2.0))
   (fma -0.0625 (sin y) (sin x))
   2.0)
  (fma
   (fma (- 1.0 (sqrt 5.0)) (cos x) (* (- (sqrt 5.0) 3.0) (cos y)))
   -1.5
   3.0)))

C

double code(double x, double y) {
	return fma(((fma(-0.0625, sin(x), sin(y)) * (cos(x) - cos(y))) * sqrt(2.0)), fma(-0.0625, sin(y), sin(x)), 2.0) / fma(fma((1.0 - sqrt(5.0)), cos(x), ((sqrt(5.0) - 3.0) * cos(y))), -1.5, 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(fma(-0.0625, sin(x), sin(y)) * Float64(cos(x) - cos(y))) * sqrt(2.0)), fma(-0.0625, sin(y), sin(x)), 2.0) / fma(fma(Float64(1.0 - sqrt(5.0)), cos(x), Float64(Float64(sqrt(5.0) - 3.0) * cos(y))), -1.5, 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

Alternative 6: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\cos y - \cos x, \sin y \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)}\\ t_1 := y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\\ \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, t\_1\right) \cdot \mathsf{fma}\left(t\_1, -0.0625, \sin x\right), 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{-2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (- (cos y) (cos x))
           (* (sin y) (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)))
           -2.0)
          (-
           (fma
            (fma (- (sqrt 5.0) 3.0) (cos y) (* (- 1.0 (sqrt 5.0)) (cos x)))
            -1.5
            3.0))))
        (t_1 (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0))))))
   (if (<= y -0.15)
     t_0
     (if (<= y 0.00068)
       (/
        (*
         (fma
          (* (- (cos x) (cos y)) (sqrt 2.0))
          (* (fma (sin x) -0.0625 t_1) (fma t_1 -0.0625 (sin x)))
          2.0)
         0.3333333333333333)
        (-
         1.0
         (/
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          -2.0)))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((cos(y) - cos(x)), (sin(y) * (fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / -fma(fma((sqrt(5.0) - 3.0), cos(y), ((1.0 - sqrt(5.0)) * cos(x))), -1.5, 3.0);
	double t_1 = y * (1.0 + (-0.16666666666666666 * pow(y, 2.0)));
	double tmp;
	if (y <= -0.15) {
		tmp = t_0;
	} else if (y <= 0.00068) {
		tmp = (fma(((cos(x) - cos(y)) * sqrt(2.0)), (fma(sin(x), -0.0625, t_1) * fma(t_1, -0.0625, sin(x))), 2.0) * 0.3333333333333333) / (1.0 - (fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))) / -2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(cos(y) - cos(x)), Float64(sin(y) * Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / Float64(-fma(fma(Float64(sqrt(5.0) - 3.0), cos(y), Float64(Float64(1.0 - sqrt(5.0)) * cos(x))), -1.5, 3.0)))
	t_1 = Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0))))
	tmp = 0.0
	if (y <= -0.15)
		tmp = t_0;
	elseif (y <= 0.00068)
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, t_1) * fma(t_1, -0.0625, sin(x))), 2.0) * 0.3333333333333333) / Float64(1.0 - Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) / -2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.15], t$95$0, If[LessEqual[y, 0.00068], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + t$95$1), $MachinePrecision] * N[(t$95$1 * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(1.0 - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.149999999999999994 or 6.8e-4 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   10. Applied rewrites 63.8%

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

   if -0.149999999999999994 < y < 6.8e-4

   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 99.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.6

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

   6. Applied rewrites 50.6%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.2

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

   9. Applied rewrites 50.2%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}, -0.0625, \sin x\right), 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{-2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)\\ t_1 := y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\\ t_2 := \cos y - \cos x\\ t_3 := \frac{\mathsf{fma}\left(t\_2, \sin y \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \mathsf{fma}\left(-0.0625, \sin x, t\_1\right) \cdot \left(\mathsf{fma}\left(-0.0625, t\_1, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (-
          (fma
           (fma (- (sqrt 5.0) 3.0) (cos y) (* (- 1.0 (sqrt 5.0)) (cos x)))
           -1.5
           3.0)))
        (t_1 (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0)))))
        (t_2 (- (cos y) (cos x)))
        (t_3
         (/
          (fma
           t_2
           (* (sin y) (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)))
           -2.0)
          t_0)))
   (if (<= y -0.15)
     t_3
     (if (<= y 0.00068)
       (/
        (fma
         t_2
         (* (fma -0.0625 (sin x) t_1) (* (fma -0.0625 t_1 (sin x)) (sqrt 2.0)))
         -2.0)
        t_0)
       t_3))))

C

double code(double x, double y) {
	double t_0 = -fma(fma((sqrt(5.0) - 3.0), cos(y), ((1.0 - sqrt(5.0)) * cos(x))), -1.5, 3.0);
	double t_1 = y * (1.0 + (-0.16666666666666666 * pow(y, 2.0)));
	double t_2 = cos(y) - cos(x);
	double t_3 = fma(t_2, (sin(y) * (fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / t_0;
	double tmp;
	if (y <= -0.15) {
		tmp = t_3;
	} else if (y <= 0.00068) {
		tmp = fma(t_2, (fma(-0.0625, sin(x), t_1) * (fma(-0.0625, t_1, sin(x)) * sqrt(2.0))), -2.0) / t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(-fma(fma(Float64(sqrt(5.0) - 3.0), cos(y), Float64(Float64(1.0 - sqrt(5.0)) * cos(x))), -1.5, 3.0))
	t_1 = Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0))))
	t_2 = Float64(cos(y) - cos(x))
	t_3 = Float64(fma(t_2, Float64(sin(y) * Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / t_0)
	tmp = 0.0
	if (y <= -0.15)
		tmp = t_3;
	elseif (y <= 0.00068)
		tmp = Float64(fma(t_2, Float64(fma(-0.0625, sin(x), t_1) * Float64(fma(-0.0625, t_1, sin(x)) * sqrt(2.0))), -2.0) / t_0);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = (-N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])}, Block[{t$95$1 = N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.15], t$95$3, If[LessEqual[y, 0.00068], N[(N[(t$95$2 * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + t$95$1), $MachinePrecision] * N[(N[(-0.0625 * t$95$1 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.149999999999999994 or 6.8e-4 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   10. Applied rewrites 63.8%

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

   if -0.149999999999999994 < y < 6.8e-4

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.6

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{\color{blue}{2}}\right)\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   10. Applied rewrites 50.6%

       \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \color{blue}{y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   11. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.1

          \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(-0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{\color{blue}{2}}\right), \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   13. Applied rewrites 50.1%

       \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(-0.0625, \color{blue}{y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 3\\ t_1 := 1 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\cos y - \cos x, \sin y \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos y, t\_1 \cdot \cos x\right), -1.5, 3\right)}\\ \mathbf{if}\;y \leq -0.011:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\left(\left(\left(\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right) - \cos x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \sqrt{2} - 1\right) - 1}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_0 \cdot \left(1 + -0.5 \cdot {y}^{2}\right)\right), -1.5, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 3.0))
        (t_1 (- 1.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (- (cos y) (cos x))
           (* (sin y) (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)))
           -2.0)
          (- (fma (fma t_0 (cos y) (* t_1 (cos x))) -1.5 3.0)))))
   (if (<= y -0.011)
     t_2
     (if (<= y 0.00068)
       (/
        (-
         (-
          (*
           (*
            (*
             (- (fma (* y y) -0.5 1.0) (cos x))
             (fma (sin x) -0.0625 (sin y)))
            (fma (sin y) -0.0625 (sin x)))
           (sqrt 2.0))
          1.0)
         1.0)
        (-
         (fma
          (fma t_1 (cos x) (* t_0 (+ 1.0 (* -0.5 (pow y 2.0)))))
          -1.5
          3.0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 3.0;
	double t_1 = 1.0 - sqrt(5.0);
	double t_2 = fma((cos(y) - cos(x)), (sin(y) * (fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / -fma(fma(t_0, cos(y), (t_1 * cos(x))), -1.5, 3.0);
	double tmp;
	if (y <= -0.011) {
		tmp = t_2;
	} else if (y <= 0.00068) {
		tmp = ((((((fma((y * y), -0.5, 1.0) - cos(x)) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))) * sqrt(2.0)) - 1.0) - 1.0) / -fma(fma(t_1, cos(x), (t_0 * (1.0 + (-0.5 * pow(y, 2.0))))), -1.5, 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 3.0)
	t_1 = Float64(1.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64(cos(y) - cos(x)), Float64(sin(y) * Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / Float64(-fma(fma(t_0, cos(y), Float64(t_1 * cos(x))), -1.5, 3.0)))
	tmp = 0.0
	if (y <= -0.011)
		tmp = t_2;
	elseif (y <= 0.00068)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(fma(Float64(y * y), -0.5, 1.0) - cos(x)) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))) * sqrt(2.0)) - 1.0) - 1.0) / Float64(-fma(fma(t_1, cos(x), Float64(t_0 * Float64(1.0 + Float64(-0.5 * (y ^ 2.0))))), -1.5, 3.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[y, -0.011], t$95$2, If[LessEqual[y, 0.00068], N[(N[(N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] - 1.0), $MachinePrecision] / (-N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[(1.0 + N[(-0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.010999999999999999 or 6.8e-4 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   10. Applied rewrites 63.8%

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

   if -0.010999999999999999 < y < 6.8e-4

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.3

         \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)} - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 52.6

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right)\right), -1.5, 3\right)} \]

   11. Applied rewrites 52.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)}\right), -1.5, 3\right)} \]
   12. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. add-flip N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. associate--r+ N/A

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

       6. lower--.f64 N/A

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

   13. Applied rewrites 52.5%

       \[\leadsto \frac{\color{blue}{\left(\left(\left(\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right) - \cos x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \sqrt{2} - 1\right) - 1}}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \left(1 + -0.5 \cdot {y}^{2}\right)\right), -1.5, 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - \sqrt{5}\right) \cdot \cos x\\ t_1 := \sqrt{5} - 3\\ t_2 := \frac{\mathsf{fma}\left(\cos y - \cos x, \sin y \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos y, t\_0\right), -1.5, 3\right)}\\ t_3 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ \mathbf{if}\;y \leq -0.011:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(t\_3 - \cos x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2}, -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_3, t\_1, t\_0\right), 1.5, -3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- 1.0 (sqrt 5.0)) (cos x)))
        (t_1 (- (sqrt 5.0) 3.0))
        (t_2
         (/
          (fma
           (- (cos y) (cos x))
           (* (sin y) (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)))
           -2.0)
          (- (fma (fma t_1 (cos y) t_0) -1.5 3.0))))
        (t_3 (fma (* y y) -0.5 1.0)))
   (if (<= y -0.011)
     t_2
     (if (<= y 0.00068)
       (/
        (fma
         (*
          (* (- t_3 (cos x)) (fma (sin x) -0.0625 (sin y)))
          (fma (sin y) -0.0625 (sin x)))
         (sqrt 2.0)
         -2.0)
        (fma (fma t_3 t_1 t_0) 1.5 -3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = (1.0 - sqrt(5.0)) * cos(x);
	double t_1 = sqrt(5.0) - 3.0;
	double t_2 = fma((cos(y) - cos(x)), (sin(y) * (fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / -fma(fma(t_1, cos(y), t_0), -1.5, 3.0);
	double t_3 = fma((y * y), -0.5, 1.0);
	double tmp;
	if (y <= -0.011) {
		tmp = t_2;
	} else if (y <= 0.00068) {
		tmp = fma((((t_3 - cos(x)) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), sqrt(2.0), -2.0) / fma(fma(t_3, t_1, t_0), 1.5, -3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - sqrt(5.0)) * cos(x))
	t_1 = Float64(sqrt(5.0) - 3.0)
	t_2 = Float64(fma(Float64(cos(y) - cos(x)), Float64(sin(y) * Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / Float64(-fma(fma(t_1, cos(y), t_0), -1.5, 3.0)))
	t_3 = fma(Float64(y * y), -0.5, 1.0)
	tmp = 0.0
	if (y <= -0.011)
		tmp = t_2;
	elseif (y <= 0.00068)
		tmp = Float64(fma(Float64(Float64(Float64(t_3 - cos(x)) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), sqrt(2.0), -2.0) / fma(fma(t_3, t_1, t_0), 1.5, -3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[y, -0.011], t$95$2, If[LessEqual[y, 0.00068], N[(N[(N[(N[(N[(t$95$3 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(t$95$3 * t$95$1 + t$95$0), $MachinePrecision] * 1.5 + -3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.010999999999999999 or 6.8e-4 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   10. Applied rewrites 63.8%

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

   if -0.010999999999999999 < y < 6.8e-4

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.3

         \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)} - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 52.6

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right)\right), -1.5, 3\right)} \]

   11. Applied rewrites 52.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)}\right), -1.5, 3\right)} \]

   13. Applied rewrites 52.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right) - \cos x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2}, -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right), \sqrt{5} - 3, \left(1 - \sqrt{5}\right) \cdot \cos x\right), 1.5, -3\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 3\\ t_1 := 1 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_0 \cdot \cos y\right), -1.5, 3\right)}\\ t_3 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ \mathbf{if}\;y \leq -0.011:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(t\_3 - \cos x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2}, -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_3, t\_0, t\_1 \cdot \cos x\right), 1.5, -3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 3.0))
        (t_1 (- 1.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (* (- (cos x) (cos y)) (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)))
           (sin y)
           2.0)
          (fma (fma t_1 (cos x) (* t_0 (cos y))) -1.5 3.0)))
        (t_3 (fma (* y y) -0.5 1.0)))
   (if (<= y -0.011)
     t_2
     (if (<= y 0.00068)
       (/
        (fma
         (*
          (* (- t_3 (cos x)) (fma (sin x) -0.0625 (sin y)))
          (fma (sin y) -0.0625 (sin x)))
         (sqrt 2.0)
         -2.0)
        (fma (fma t_3 t_0 (* t_1 (cos x))) 1.5 -3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 3.0;
	double t_1 = 1.0 - sqrt(5.0);
	double t_2 = fma(((cos(x) - cos(y)) * (fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), sin(y), 2.0) / fma(fma(t_1, cos(x), (t_0 * cos(y))), -1.5, 3.0);
	double t_3 = fma((y * y), -0.5, 1.0);
	double tmp;
	if (y <= -0.011) {
		tmp = t_2;
	} else if (y <= 0.00068) {
		tmp = fma((((t_3 - cos(x)) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), sqrt(2.0), -2.0) / fma(fma(t_3, t_0, (t_1 * cos(x))), 1.5, -3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 3.0)
	t_1 = Float64(1.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64(Float64(cos(x) - cos(y)) * Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), sin(y), 2.0) / fma(fma(t_1, cos(x), Float64(t_0 * cos(y))), -1.5, 3.0))
	t_3 = fma(Float64(y * y), -0.5, 1.0)
	tmp = 0.0
	if (y <= -0.011)
		tmp = t_2;
	elseif (y <= 0.00068)
		tmp = Float64(fma(Float64(Float64(Float64(t_3 - cos(x)) * fma(sin(x), -0.0625, sin(y))) * fma(sin(y), -0.0625, sin(x))), sqrt(2.0), -2.0) / fma(fma(t_3, t_0, Float64(t_1 * cos(x))), 1.5, -3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[y, -0.011], t$95$2, If[LessEqual[y, 0.00068], N[(N[(N[(N[(N[(t$95$3 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(t$95$3 * t$95$0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + -3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.010999999999999999 or 6.8e-4 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 63.8%

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

   if -0.010999999999999999 < y < 6.8e-4

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.3

         \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)} - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 52.6

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right)\right), -1.5, 3\right)} \]

   11. Applied rewrites 52.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)}\right), -1.5, 3\right)} \]

   13. Applied rewrites 52.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right) - \cos x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2}, -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right), \sqrt{5} - 3, \left(1 - \sqrt{5}\right) \cdot \cos x\right), 1.5, -3\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_2 := 1 - \sqrt{5}\\ t_3 := \sqrt{5} - 3\\ t_4 := t\_2 \cdot \cos x\\ \mathbf{if}\;y \leq -0.011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, t\_0, 2\right)}{\mathsf{fma}\left(-1.5, \mathsf{fma}\left(t\_2, \cos x, t\_3 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(t\_1 - \cos x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot t\_0, \sqrt{2}, -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, t\_3, t\_4\right), 1.5, -3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos y - \cos x, -0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_3, \cos y, t\_4\right), -1.5, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (fma (* y y) -0.5 1.0))
        (t_2 (- 1.0 (sqrt 5.0)))
        (t_3 (- (sqrt 5.0) 3.0))
        (t_4 (* t_2 (cos x))))
   (if (<= y -0.011)
     (/
      (fma (* (* (- 1.0 (cos y)) (sqrt 2.0)) (sin y)) t_0 2.0)
      (fma -1.5 (fma t_2 (cos x) (* t_3 (cos y))) 3.0))
     (if (<= y 0.00068)
       (/
        (fma
         (* (* (- t_1 (cos x)) (fma (sin x) -0.0625 (sin y))) t_0)
         (sqrt 2.0)
         -2.0)
        (fma (fma t_1 t_3 t_4) 1.5 -3.0))
       (/
        (fma
         (- (cos y) (cos x))
         (* -0.0625 (* (pow (sin y) 2.0) (sqrt 2.0)))
         -2.0)
        (- (fma (fma t_3 (cos y) t_4) -1.5 3.0)))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = fma((y * y), -0.5, 1.0);
	double t_2 = 1.0 - sqrt(5.0);
	double t_3 = sqrt(5.0) - 3.0;
	double t_4 = t_2 * cos(x);
	double tmp;
	if (y <= -0.011) {
		tmp = fma((((1.0 - cos(y)) * sqrt(2.0)) * sin(y)), t_0, 2.0) / fma(-1.5, fma(t_2, cos(x), (t_3 * cos(y))), 3.0);
	} else if (y <= 0.00068) {
		tmp = fma((((t_1 - cos(x)) * fma(sin(x), -0.0625, sin(y))) * t_0), sqrt(2.0), -2.0) / fma(fma(t_1, t_3, t_4), 1.5, -3.0);
	} else {
		tmp = fma((cos(y) - cos(x)), (-0.0625 * (pow(sin(y), 2.0) * sqrt(2.0))), -2.0) / -fma(fma(t_3, cos(y), t_4), -1.5, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = fma(Float64(y * y), -0.5, 1.0)
	t_2 = Float64(1.0 - sqrt(5.0))
	t_3 = Float64(sqrt(5.0) - 3.0)
	t_4 = Float64(t_2 * cos(x))
	tmp = 0.0
	if (y <= -0.011)
		tmp = Float64(fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * sin(y)), t_0, 2.0) / fma(-1.5, fma(t_2, cos(x), Float64(t_3 * cos(y))), 3.0));
	elseif (y <= 0.00068)
		tmp = Float64(fma(Float64(Float64(Float64(t_1 - cos(x)) * fma(sin(x), -0.0625, sin(y))) * t_0), sqrt(2.0), -2.0) / fma(fma(t_1, t_3, t_4), 1.5, -3.0));
	else
		tmp = Float64(fma(Float64(cos(y) - cos(x)), Float64(-0.0625 * Float64((sin(y) ^ 2.0) * sqrt(2.0))), -2.0) / Float64(-fma(fma(t_3, cos(y), t_4), -1.5, 3.0)));
	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[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.011], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(-1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00068], N[(N[(N[(N[(N[(t$95$1 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(t$95$1 * t$95$3 + t$95$4), $MachinePrecision] * 1.5 + -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$4), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.010999999999999999

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   6. Applied rewrites 62.1%

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

   8. Applied rewrites 62.1%

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

   if -0.010999999999999999 < y < 6.8e-4

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.3

         \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)} - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 52.6

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right)\right), -1.5, 3\right)} \]

   11. Applied rewrites 52.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)}\right), -1.5, 3\right)} \]

   13. Applied rewrites 52.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right) - \cos x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2}, -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right), \sqrt{5} - 3, \left(1 - \sqrt{5}\right) \cdot \cos x\right), 1.5, -3\right)}} \]

   if 6.8e-4 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.1

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

   10. Applied rewrites 62.1%

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

Alternative 12: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + -0.5 \cdot {y}^{2}\\ t_1 := 1 - \sqrt{5}\\ t_2 := \sqrt{5} - 3\\ \mathbf{if}\;y \leq -0.011:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(-1.5, \mathsf{fma}\left(t\_1, \cos x, t\_2 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\sin x + -0.0625 \cdot y\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_2 \cdot t\_0\right), -1.5, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos y - \cos x, -0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos y, t\_1 \cdot \cos x\right), -1.5, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -0.5 (pow y 2.0))))
        (t_1 (- 1.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 3.0)))
   (if (<= y -0.011)
     (/
      (fma
       (* (* (- 1.0 (cos y)) (sqrt 2.0)) (sin y))
       (fma (sin y) -0.0625 (sin x))
       2.0)
      (fma -1.5 (fma t_1 (cos x) (* t_2 (cos y))) 3.0))
     (if (<= y 0.00068)
       (/
        (fma
         (- t_0 (cos x))
         (*
          (fma -0.0625 (sin x) (sin y))
          (* (+ (sin x) (* -0.0625 y)) (sqrt 2.0)))
         -2.0)
        (- (fma (fma t_1 (cos x) (* t_2 t_0)) -1.5 3.0)))
       (/
        (fma
         (- (cos y) (cos x))
         (* -0.0625 (* (pow (sin y) 2.0) (sqrt 2.0)))
         -2.0)
        (- (fma (fma t_2 (cos y) (* t_1 (cos x))) -1.5 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-0.5 * pow(y, 2.0));
	double t_1 = 1.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 3.0;
	double tmp;
	if (y <= -0.011) {
		tmp = fma((((1.0 - cos(y)) * sqrt(2.0)) * sin(y)), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(-1.5, fma(t_1, cos(x), (t_2 * cos(y))), 3.0);
	} else if (y <= 0.00068) {
		tmp = fma((t_0 - cos(x)), (fma(-0.0625, sin(x), sin(y)) * ((sin(x) + (-0.0625 * y)) * sqrt(2.0))), -2.0) / -fma(fma(t_1, cos(x), (t_2 * t_0)), -1.5, 3.0);
	} else {
		tmp = fma((cos(y) - cos(x)), (-0.0625 * (pow(sin(y), 2.0) * sqrt(2.0))), -2.0) / -fma(fma(t_2, cos(y), (t_1 * cos(x))), -1.5, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-0.5 * (y ^ 2.0)))
	t_1 = Float64(1.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 3.0)
	tmp = 0.0
	if (y <= -0.011)
		tmp = Float64(fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * sin(y)), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(-1.5, fma(t_1, cos(x), Float64(t_2 * cos(y))), 3.0));
	elseif (y <= 0.00068)
		tmp = Float64(fma(Float64(t_0 - cos(x)), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(Float64(sin(x) + Float64(-0.0625 * y)) * sqrt(2.0))), -2.0) / Float64(-fma(fma(t_1, cos(x), Float64(t_2 * t_0)), -1.5, 3.0)));
	else
		tmp = Float64(fma(Float64(cos(y) - cos(x)), Float64(-0.0625 * Float64((sin(y) ^ 2.0) * sqrt(2.0))), -2.0) / Float64(-fma(fma(t_2, cos(y), Float64(t_1 * cos(x))), -1.5, 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, If[LessEqual[y, -0.011], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * 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[(-1.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00068], N[(N[(N[(t$95$0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision], N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.010999999999999999

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   6. Applied rewrites 62.1%

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

   8. Applied rewrites 62.1%

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

   if -0.010999999999999999 < y < 6.8e-4

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.3

         \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)} - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 52.6

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right)\right), -1.5, 3\right)} \]

   11. Applied rewrites 52.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)}\right), -1.5, 3\right)} \]

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 50.3

          \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(\sin x + -0.0625 \cdot \color{blue}{y}\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \left(1 + -0.5 \cdot {y}^{2}\right)\right), -1.5, 3\right)} \]

   14. Applied rewrites 50.3%

       \[\leadsto \frac{\mathsf{fma}\left(\left(1 + -0.5 \cdot {y}^{2}\right) - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(\sin x + -0.0625 \cdot y\right)} \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \left(1 + -0.5 \cdot {y}^{2}\right)\right), -1.5, 3\right)} \]

   if 6.8e-4 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.1

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

   10. Applied rewrites 62.1%

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

Alternative 13: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \sqrt{5}\\ t_1 := \sqrt{5} - 3\\ \mathbf{if}\;y \leq -0.0136:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(-1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\cos y, t\_1, \cos x \cdot t\_0\right)}{-2} \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos y - \cos x, -0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos y, t\_0 \cdot \cos x\right), -1.5, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (sqrt 5.0))) (t_1 (- (sqrt 5.0) 3.0)))
   (if (<= y -0.0136)
     (/
      (fma
       (* (* (- 1.0 (cos y)) (sqrt 2.0)) (sin y))
       (fma (sin y) -0.0625 (sin x))
       2.0)
      (fma -1.5 (fma t_0 (cos x) (* t_1 (cos y))) 3.0))
     (if (<= y 0.00068)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          (- (cos x) (cos y))))
        (+ 3.0 (* (/ (fma (cos y) t_1 (* (cos x) t_0)) -2.0) 3.0)))
       (/
        (fma
         (- (cos y) (cos x))
         (* -0.0625 (* (pow (sin y) 2.0) (sqrt 2.0)))
         -2.0)
        (- (fma (fma t_1 (cos y) (* t_0 (cos x))) -1.5 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 3.0;
	double tmp;
	if (y <= -0.0136) {
		tmp = fma((((1.0 - cos(y)) * sqrt(2.0)) * sin(y)), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(-1.5, fma(t_0, cos(x), (t_1 * cos(y))), 3.0);
	} else if (y <= 0.00068) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 + ((fma(cos(y), t_1, (cos(x) * t_0)) / -2.0) * 3.0));
	} else {
		tmp = fma((cos(y) - cos(x)), (-0.0625 * (pow(sin(y), 2.0) * sqrt(2.0))), -2.0) / -fma(fma(t_1, cos(y), (t_0 * cos(x))), -1.5, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 3.0)
	tmp = 0.0
	if (y <= -0.0136)
		tmp = Float64(fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * sin(y)), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(-1.5, fma(t_0, cos(x), Float64(t_1 * cos(y))), 3.0));
	elseif (y <= 0.00068)
		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))) * Float64(cos(x) - cos(y)))) / Float64(3.0 + Float64(Float64(fma(cos(y), t_1, Float64(cos(x) * t_0)) / -2.0) * 3.0)));
	else
		tmp = Float64(fma(Float64(cos(y) - cos(x)), Float64(-0.0625 * Float64((sin(y) ^ 2.0) * sqrt(2.0))), -2.0) / Float64(-fma(fma(t_1, cos(y), Float64(t_0 * cos(x))), -1.5, 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, If[LessEqual[y, -0.0136], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * 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[(-1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00068], 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] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0135999999999999992

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   6. Applied rewrites 62.1%

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

   8. Applied rewrites 62.1%

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

   if -0.0135999999999999992 < y < 6.8e-4

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 51.1%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 50.2%

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

   if 6.8e-4 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.1

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

   10. Applied rewrites 62.1%

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

Alternative 14: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \sqrt{5}\\ t_1 := \sqrt{5} - 3\\ \mathbf{if}\;y \leq -0.0136:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(-1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, y\right) \cdot \mathsf{fma}\left(y, -0.0625, \sin x\right), 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{-2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos y - \cos x, -0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos y, t\_0 \cdot \cos x\right), -1.5, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (sqrt 5.0))) (t_1 (- (sqrt 5.0) 3.0)))
   (if (<= y -0.0136)
     (/
      (fma
       (* (* (- 1.0 (cos y)) (sqrt 2.0)) (sin y))
       (fma (sin y) -0.0625 (sin x))
       2.0)
      (fma -1.5 (fma t_0 (cos x) (* t_1 (cos y))) 3.0))
     (if (<= y 0.00068)
       (/
        (*
         (fma
          (* (- (cos x) (cos y)) (sqrt 2.0))
          (* (fma (sin x) -0.0625 y) (fma y -0.0625 (sin x)))
          2.0)
         0.3333333333333333)
        (-
         1.0
         (/
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          -2.0)))
       (/
        (fma
         (- (cos y) (cos x))
         (* -0.0625 (* (pow (sin y) 2.0) (sqrt 2.0)))
         -2.0)
        (- (fma (fma t_1 (cos y) (* t_0 (cos x))) -1.5 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 3.0;
	double tmp;
	if (y <= -0.0136) {
		tmp = fma((((1.0 - cos(y)) * sqrt(2.0)) * sin(y)), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(-1.5, fma(t_0, cos(x), (t_1 * cos(y))), 3.0);
	} else if (y <= 0.00068) {
		tmp = (fma(((cos(x) - cos(y)) * sqrt(2.0)), (fma(sin(x), -0.0625, y) * fma(y, -0.0625, sin(x))), 2.0) * 0.3333333333333333) / (1.0 - (fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))) / -2.0));
	} else {
		tmp = fma((cos(y) - cos(x)), (-0.0625 * (pow(sin(y), 2.0) * sqrt(2.0))), -2.0) / -fma(fma(t_1, cos(y), (t_0 * cos(x))), -1.5, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 3.0)
	tmp = 0.0
	if (y <= -0.0136)
		tmp = Float64(fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * sin(y)), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(-1.5, fma(t_0, cos(x), Float64(t_1 * cos(y))), 3.0));
	elseif (y <= 0.00068)
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, y) * fma(y, -0.0625, sin(x))), 2.0) * 0.3333333333333333) / Float64(1.0 - Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) / -2.0)));
	else
		tmp = Float64(fma(Float64(cos(y) - cos(x)), Float64(-0.0625 * Float64((sin(y) ^ 2.0) * sqrt(2.0))), -2.0) / Float64(-fma(fma(t_1, cos(y), Float64(t_0 * cos(x))), -1.5, 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, If[LessEqual[y, -0.0136], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * 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[(-1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00068], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + y), $MachinePrecision] * N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(1.0 - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0135999999999999992

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   6. Applied rewrites 62.1%

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

   8. Applied rewrites 62.1%

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

   if -0.0135999999999999992 < y < 6.8e-4

   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 99.2%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 50.3%

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

   if 6.8e-4 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.1

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

   10. Applied rewrites 62.1%

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

Alternative 15: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 3\\ t_2 := {\sin x}^{2}\\ t_3 := 1 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0002:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) \cdot t\_0}{3 + \frac{\mathsf{fma}\left(\cos y, t\_1, \cos x \cdot t\_3\right)}{-2} \cdot 3}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos y - 1, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos y, t\_3 \cdot 1\right), -1.5, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, -0.0625 \cdot t\_2, 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 3.0))
        (t_2 (pow (sin x) 2.0))
        (t_3 (- 1.0 (sqrt 5.0))))
   (if (<= x -0.0002)
     (/
      (+ 2.0 (* (* -0.0625 (* t_2 (sqrt 2.0))) t_0))
      (+ 3.0 (* (/ (fma (cos y) t_1 (* (cos x) t_3)) -2.0) 3.0)))
     (if (<= x 2.3e-5)
       (/
        (fma
         (- (cos y) 1.0)
         (*
          (fma -0.0625 (sin x) (sin y))
          (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)))
         -2.0)
        (- (fma (fma t_1 (cos y) (* t_3 1.0)) -1.5 3.0)))
       (/
        (* (fma (* t_0 (sqrt 2.0)) (* -0.0625 t_2) 2.0) 0.3333333333333333)
        (-
         1.0
         (/
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          -2.0)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 3.0;
	double t_2 = pow(sin(x), 2.0);
	double t_3 = 1.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.0002) {
		tmp = (2.0 + ((-0.0625 * (t_2 * sqrt(2.0))) * t_0)) / (3.0 + ((fma(cos(y), t_1, (cos(x) * t_3)) / -2.0) * 3.0));
	} else if (x <= 2.3e-5) {
		tmp = fma((cos(y) - 1.0), (fma(-0.0625, sin(x), sin(y)) * (fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / -fma(fma(t_1, cos(y), (t_3 * 1.0)), -1.5, 3.0);
	} else {
		tmp = (fma((t_0 * sqrt(2.0)), (-0.0625 * t_2), 2.0) * 0.3333333333333333) / (1.0 - (fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))) / -2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 3.0)
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(1.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0002)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_2 * sqrt(2.0))) * t_0)) / Float64(3.0 + Float64(Float64(fma(cos(y), t_1, Float64(cos(x) * t_3)) / -2.0) * 3.0)));
	elseif (x <= 2.3e-5)
		tmp = Float64(fma(Float64(cos(y) - 1.0), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / Float64(-fma(fma(t_1, cos(y), Float64(t_3 * 1.0)), -1.5, 3.0)));
	else
		tmp = Float64(Float64(fma(Float64(t_0 * sqrt(2.0)), Float64(-0.0625 * t_2), 2.0) * 0.3333333333333333) / Float64(1.0 - Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) / -2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0002], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-5], N[(N[(N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(t$95$3 * 1.0), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(1.0 - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e-4

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3

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

   6. Applied rewrites 62.3%

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

   if -2.0000000000000001e-4 < x < 2.3e-5

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.2%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 59.5%

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

   if 2.3e-5 < 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 99.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.3

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

   6. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{-2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 3\\ t_2 := {\sin x}^{2}\\ t_3 := 1 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0002:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) \cdot t\_0}{3 + \frac{\mathsf{fma}\left(\cos y, t\_1, \cos x \cdot t\_3\right)}{-2} \cdot 3}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos y - 1, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_3, 1, t\_1 \cdot \cos y\right), -1.5, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, -0.0625 \cdot t\_2, 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 3.0))
        (t_2 (pow (sin x) 2.0))
        (t_3 (- 1.0 (sqrt 5.0))))
   (if (<= x -0.0002)
     (/
      (+ 2.0 (* (* -0.0625 (* t_2 (sqrt 2.0))) t_0))
      (+ 3.0 (* (/ (fma (cos y) t_1 (* (cos x) t_3)) -2.0) 3.0)))
     (if (<= x 2.3e-5)
       (/
        (fma
         (- (cos y) 1.0)
         (*
          (fma -0.0625 (sin x) (sin y))
          (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)))
         -2.0)
        (- (fma (fma t_3 1.0 (* t_1 (cos y))) -1.5 3.0)))
       (/
        (* (fma (* t_0 (sqrt 2.0)) (* -0.0625 t_2) 2.0) 0.3333333333333333)
        (-
         1.0
         (/
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          -2.0)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 3.0;
	double t_2 = pow(sin(x), 2.0);
	double t_3 = 1.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.0002) {
		tmp = (2.0 + ((-0.0625 * (t_2 * sqrt(2.0))) * t_0)) / (3.0 + ((fma(cos(y), t_1, (cos(x) * t_3)) / -2.0) * 3.0));
	} else if (x <= 2.3e-5) {
		tmp = fma((cos(y) - 1.0), (fma(-0.0625, sin(x), sin(y)) * (fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / -fma(fma(t_3, 1.0, (t_1 * cos(y))), -1.5, 3.0);
	} else {
		tmp = (fma((t_0 * sqrt(2.0)), (-0.0625 * t_2), 2.0) * 0.3333333333333333) / (1.0 - (fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))) / -2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 3.0)
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(1.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0002)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_2 * sqrt(2.0))) * t_0)) / Float64(3.0 + Float64(Float64(fma(cos(y), t_1, Float64(cos(x) * t_3)) / -2.0) * 3.0)));
	elseif (x <= 2.3e-5)
		tmp = Float64(fma(Float64(cos(y) - 1.0), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0))), -2.0) / Float64(-fma(fma(t_3, 1.0, Float64(t_1 * cos(y))), -1.5, 3.0)));
	else
		tmp = Float64(Float64(fma(Float64(t_0 * sqrt(2.0)), Float64(-0.0625 * t_2), 2.0) * 0.3333333333333333) / Float64(1.0 - Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) / -2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0002], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-5], N[(N[(N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(t$95$3 * 1.0 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(1.0 - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e-4

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3

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

   6. Applied rewrites 62.3%

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

   if -2.0000000000000001e-4 < x < 2.3e-5

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 59.5%

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

   if 2.3e-5 < 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 99.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.3

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

   6. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{-2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 3\\ t_2 := {\sin x}^{2}\\ t_3 := 1 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0029:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) \cdot t\_0}{3 + \frac{\mathsf{fma}\left(\cos y, t\_1, \cos x \cdot t\_3\right)}{-2} \cdot 3}\\ \mathbf{elif}\;x \leq 0.0076:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(-1.5, \mathsf{fma}\left(t\_3, \cos x, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, -0.0625 \cdot t\_2, 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 3.0))
        (t_2 (pow (sin x) 2.0))
        (t_3 (- 1.0 (sqrt 5.0))))
   (if (<= x -0.0029)
     (/
      (+ 2.0 (* (* -0.0625 (* t_2 (sqrt 2.0))) t_0))
      (+ 3.0 (* (/ (fma (cos y) t_1 (* (cos x) t_3)) -2.0) 3.0)))
     (if (<= x 0.0076)
       (/
        (fma
         (* (* (- 1.0 (cos y)) (sqrt 2.0)) (sin y))
         (fma (sin y) -0.0625 (sin x))
         2.0)
        (fma -1.5 (fma t_3 (cos x) (* t_1 (cos y))) 3.0))
       (/
        (* (fma (* t_0 (sqrt 2.0)) (* -0.0625 t_2) 2.0) 0.3333333333333333)
        (-
         1.0
         (/
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          -2.0)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 3.0;
	double t_2 = pow(sin(x), 2.0);
	double t_3 = 1.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.0029) {
		tmp = (2.0 + ((-0.0625 * (t_2 * sqrt(2.0))) * t_0)) / (3.0 + ((fma(cos(y), t_1, (cos(x) * t_3)) / -2.0) * 3.0));
	} else if (x <= 0.0076) {
		tmp = fma((((1.0 - cos(y)) * sqrt(2.0)) * sin(y)), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(-1.5, fma(t_3, cos(x), (t_1 * cos(y))), 3.0);
	} else {
		tmp = (fma((t_0 * sqrt(2.0)), (-0.0625 * t_2), 2.0) * 0.3333333333333333) / (1.0 - (fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))) / -2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 3.0)
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(1.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0029)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_2 * sqrt(2.0))) * t_0)) / Float64(3.0 + Float64(Float64(fma(cos(y), t_1, Float64(cos(x) * t_3)) / -2.0) * 3.0)));
	elseif (x <= 0.0076)
		tmp = Float64(fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * sin(y)), fma(sin(y), -0.0625, sin(x)), 2.0) / fma(-1.5, fma(t_3, cos(x), Float64(t_1 * cos(y))), 3.0));
	else
		tmp = Float64(Float64(fma(Float64(t_0 * sqrt(2.0)), Float64(-0.0625 * t_2), 2.0) * 0.3333333333333333) / Float64(1.0 - Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) / -2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0029], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0076], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * 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[(-1.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(1.0 - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0029

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3

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

   6. Applied rewrites 62.3%

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

   if -0.0029 < x < 0.00759999999999999998

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   6. Applied rewrites 62.1%

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

   8. Applied rewrites 62.1%

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

   if 0.00759999999999999998 < 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 99.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.3

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

   6. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{-2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin x}^{2}\\ t_2 := \cos x - \cos y\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot t\_2}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos y \cdot t\_3, 0.5 \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, -0.0625 \cdot t\_1, 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(t\_0, \cos x, t\_3 \cdot \cos y\right)}{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -3.7e-6)
     (/
      (+ 2.0 (* (* -0.0625 (* t_1 (sqrt 2.0))) t_2))
      (+
       3.0
       (*
        (/
         (fma (cos y) (- (sqrt 5.0) 3.0) (* (cos x) (- 1.0 (sqrt 5.0))))
         -2.0)
        3.0)))
     (if (<= x 1.12e-5)
       (/
        (+
         2.0
         (*
          (fma (sin y) -0.0625 (sin x))
          (* (sin y) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (* 3.0 (+ 1.0 (fma 0.5 (* (cos y) t_3) (* 0.5 t_0)))))
       (/
        (* (fma (* t_2 (sqrt 2.0)) (* -0.0625 t_1) 2.0) 0.3333333333333333)
        (- 1.0 (/ (fma t_0 (cos x) (* t_3 (cos y))) -2.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = pow(sin(x), 2.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -3.7e-6) {
		tmp = (2.0 + ((-0.0625 * (t_1 * sqrt(2.0))) * t_2)) / (3.0 + ((fma(cos(y), (sqrt(5.0) - 3.0), (cos(x) * (1.0 - sqrt(5.0)))) / -2.0) * 3.0));
	} else if (x <= 1.12e-5) {
		tmp = (2.0 + (fma(sin(y), -0.0625, sin(x)) * (sin(y) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (1.0 + fma(0.5, (cos(y) * t_3), (0.5 * t_0))));
	} else {
		tmp = (fma((t_2 * sqrt(2.0)), (-0.0625 * t_1), 2.0) * 0.3333333333333333) / (1.0 - (fma(t_0, cos(x), (t_3 * cos(y))) / -2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -3.7e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_1 * sqrt(2.0))) * t_2)) / Float64(3.0 + Float64(Float64(fma(cos(y), Float64(sqrt(5.0) - 3.0), Float64(cos(x) * Float64(1.0 - sqrt(5.0)))) / -2.0) * 3.0)));
	elseif (x <= 1.12e-5)
		tmp = Float64(Float64(2.0 + Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(y) * t_3), Float64(0.5 * t_0)))));
	else
		tmp = Float64(Float64(fma(Float64(t_2 * sqrt(2.0)), Float64(-0.0625 * t_1), 2.0) * 0.3333333333333333) / Float64(1.0 - Float64(fma(t_0, cos(x), Float64(t_3 * cos(y))) / -2.0)));
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e-6], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e-5], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$3), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(1.0 - N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7000000000000002e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3

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

   6. Applied rewrites 62.3%

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

   if -3.7000000000000002e-6 < x < 1.11999999999999995e-5

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   6. Applied rewrites 62.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 59.4

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

   9. Applied rewrites 59.4%

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

   if 1.11999999999999995e-5 < 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 99.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.3

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

   6. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{-2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y, -1.5, \left(\left(1 - \sqrt{5}\right) \cdot \cos x\right) \cdot -1.5\right)}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos y \cdot t\_0, 0.5 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot t\_2, 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(t\_1, \cos x, t\_0 \cdot \cos y\right)}{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (pow (sin x) 2.0)))
   (if (<= x -3.7e-6)
     (/
      (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) (- (cos x) 1.0)))))
      (+
       3.0
       (fma
        (* (- (sqrt 5.0) 3.0) (cos y))
        -1.5
        (* (* (- 1.0 (sqrt 5.0)) (cos x)) -1.5))))
     (if (<= x 1.12e-5)
       (/
        (+
         2.0
         (*
          (fma (sin y) -0.0625 (sin x))
          (* (sin y) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (* 3.0 (+ 1.0 (fma 0.5 (* (cos y) t_0) (* 0.5 t_1)))))
       (/
        (*
         (fma (* (- (cos x) (cos y)) (sqrt 2.0)) (* -0.0625 t_2) 2.0)
         0.3333333333333333)
        (- 1.0 (/ (fma t_1 (cos x) (* t_0 (cos y))) -2.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -3.7e-6) {
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + fma(((sqrt(5.0) - 3.0) * cos(y)), -1.5, (((1.0 - sqrt(5.0)) * cos(x)) * -1.5)));
	} else if (x <= 1.12e-5) {
		tmp = (2.0 + (fma(sin(y), -0.0625, sin(x)) * (sin(y) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (1.0 + fma(0.5, (cos(y) * t_0), (0.5 * t_1))));
	} else {
		tmp = (fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * t_2), 2.0) * 0.3333333333333333) / (1.0 - (fma(t_1, cos(x), (t_0 * cos(y))) / -2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -3.7e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + fma(Float64(Float64(sqrt(5.0) - 3.0) * cos(y)), -1.5, Float64(Float64(Float64(1.0 - sqrt(5.0)) * cos(x)) * -1.5))));
	elseif (x <= 1.12e-5)
		tmp = Float64(Float64(2.0 + Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(y) * t_0), Float64(0.5 * t_1)))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * t_2), 2.0) * 0.3333333333333333) / Float64(1.0 - Float64(fma(t_1, cos(x), Float64(t_0 * cos(y))) / -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[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -3.7e-6], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] * -1.5 + N[(N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e-5], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(1.0 - N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7000000000000002e-6

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 62.3

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

   4. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

   6. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 62.3

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

   8. Applied rewrites 62.3%

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

   if -3.7000000000000002e-6 < x < 1.11999999999999995e-5

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   6. Applied rewrites 62.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 59.4

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

   9. Applied rewrites 59.4%

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

   if 1.11999999999999995e-5 < 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 99.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.3

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

   6. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin x}^{2}}, 2\right) \cdot 0.3333333333333333}{1 - \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{-2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{5} - 3\right) \cdot \cos y\\ t_1 := {\sin x}^{2}\\ t_2 := 1 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \mathsf{fma}\left(t\_0, -1.5, \left(t\_2 \cdot \cos x\right) \cdot -1.5\right)}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos y - \cos x, -0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, t\_0\right), -1.5, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (sqrt 5.0) 3.0) (cos y)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- 1.0 (sqrt 5.0))))
   (if (<= x -3.7e-6)
     (/
      (+ 2.0 (* -0.0625 (* t_1 (* (sqrt 2.0) (- (cos x) 1.0)))))
      (+ 3.0 (fma t_0 -1.5 (* (* t_2 (cos x)) -1.5))))
     (if (<= x 1.12e-5)
       (/
        (+
         2.0
         (*
          (fma (sin y) -0.0625 (sin x))
          (* (sin y) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (*
         3.0
         (+
          1.0
          (fma
           0.5
           (* (cos y) (- 3.0 (sqrt 5.0)))
           (* 0.5 (- (sqrt 5.0) 1.0))))))
       (/
        (fma (- (cos y) (cos x)) (* -0.0625 (* t_1 (sqrt 2.0))) -2.0)
        (- (fma (fma t_2 (cos x) t_0) -1.5 3.0)))))))

C

double code(double x, double y) {
	double t_0 = (sqrt(5.0) - 3.0) * cos(y);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = 1.0 - sqrt(5.0);
	double tmp;
	if (x <= -3.7e-6) {
		tmp = (2.0 + (-0.0625 * (t_1 * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + fma(t_0, -1.5, ((t_2 * cos(x)) * -1.5)));
	} else if (x <= 1.12e-5) {
		tmp = (2.0 + (fma(sin(y), -0.0625, sin(x)) * (sin(y) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (1.0 + fma(0.5, (cos(y) * (3.0 - sqrt(5.0))), (0.5 * (sqrt(5.0) - 1.0)))));
	} else {
		tmp = fma((cos(y) - cos(x)), (-0.0625 * (t_1 * sqrt(2.0))), -2.0) / -fma(fma(t_2, cos(x), t_0), -1.5, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(5.0) - 3.0) * cos(y))
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(1.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -3.7e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_1 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + fma(t_0, -1.5, Float64(Float64(t_2 * cos(x)) * -1.5))));
	elseif (x <= 1.12e-5)
		tmp = Float64(Float64(2.0 + Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(y) * Float64(3.0 - sqrt(5.0))), Float64(0.5 * Float64(sqrt(5.0) - 1.0))))));
	else
		tmp = Float64(fma(Float64(cos(y) - cos(x)), Float64(-0.0625 * Float64(t_1 * sqrt(2.0))), -2.0) / Float64(-fma(fma(t_2, cos(x), t_0), -1.5, 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e-6], N[(N[(2.0 + N[(-0.0625 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(t$95$0 * -1.5 + N[(N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e-5], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / (-N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7000000000000002e-6

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 62.3

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

   4. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

   6. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 62.3

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

   8. Applied rewrites 62.3%

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

   if -3.7000000000000002e-6 < x < 1.11999999999999995e-5

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   6. Applied rewrites 62.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 59.4

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

   9. Applied rewrites 59.4%

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

   if 1.11999999999999995e-5 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3

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

   8. Applied rewrites 62.3%

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

Alternative 21: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \left(\sqrt{5} - 3\right) \cdot \cos y\\ t_2 := 1 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{3 + \mathsf{fma}\left(t\_1, -1.5, \left(t\_2 \cdot \cos x\right) \cdot -1.5\right)}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-0.0625 \cdot \left(t\_0 \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_2, \cos x, t\_1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (* (- (sqrt 5.0) 3.0) (cos y)))
        (t_2 (- 1.0 (sqrt 5.0))))
   (if (<= x -3.7e-6)
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_0))))
      (+ 3.0 (fma t_1 -1.5 (* (* t_2 (cos x)) -1.5))))
     (if (<= x 1.12e-5)
       (/
        (+
         2.0
         (*
          (fma (sin y) -0.0625 (sin x))
          (* (sin y) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (*
         3.0
         (+
          1.0
          (fma
           0.5
           (* (cos y) (- 3.0 (sqrt 5.0)))
           (* 0.5 (- (sqrt 5.0) 1.0))))))
       (/
        (*
         (-
          (* (* -0.0625 (* t_0 (sqrt 2.0))) (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          -2.0)
         0.3333333333333333)
        (fma -0.5 (fma t_2 (cos x) t_1) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = (sqrt(5.0) - 3.0) * cos(y);
	double t_2 = 1.0 - sqrt(5.0);
	double tmp;
	if (x <= -3.7e-6) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_0)))) / (3.0 + fma(t_1, -1.5, ((t_2 * cos(x)) * -1.5)));
	} else if (x <= 1.12e-5) {
		tmp = (2.0 + (fma(sin(y), -0.0625, sin(x)) * (sin(y) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (1.0 + fma(0.5, (cos(y) * (3.0 - sqrt(5.0))), (0.5 * (sqrt(5.0) - 1.0)))));
	} else {
		tmp = ((((-0.0625 * (t_0 * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - -2.0) * 0.3333333333333333) / fma(-0.5, fma(t_2, cos(x), t_1), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(Float64(sqrt(5.0) - 3.0) * cos(y))
	t_2 = Float64(1.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -3.7e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_0)))) / Float64(3.0 + fma(t_1, -1.5, Float64(Float64(t_2 * cos(x)) * -1.5))));
	elseif (x <= 1.12e-5)
		tmp = Float64(Float64(2.0 + Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(y) * Float64(3.0 - sqrt(5.0))), Float64(0.5 * Float64(sqrt(5.0) - 1.0))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.0625 * Float64(t_0 * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) - -2.0) * 0.3333333333333333) / fma(-0.5, fma(t_2, cos(x), t_1), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e-6], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(t$95$1 * -1.5 + N[(N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e-5], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.0625 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7000000000000002e-6

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 62.3

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. Applied rewrites 62.3%

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \left(-0.5 \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)\right) \cdot 3}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)\right) \cdot 3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{3 \cdot \left(\frac{-1}{2} \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 3 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\left(3 \cdot \frac{-1}{2}\right) \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\frac{-3}{2}} \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{-3}{2} \cdot \color{blue}{\left(\left(1 - \sqrt{5}\right) \cdot \cos x + \left(\sqrt{5} - 3\right) \cdot \cos y\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{-3}{2} \cdot \left(\color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x} + \left(\sqrt{5} - 3\right) \cdot \cos y\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{-3}{2} \cdot \color{blue}{\left(\left(\sqrt{5} - 3\right) \cdot \cos y + \left(1 - \sqrt{5}\right) \cdot \cos x\right)}} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\left(\left(\left(\sqrt{5} - 3\right) \cdot \cos y\right) \cdot \frac{-3}{2} + \left(\left(1 - \sqrt{5}\right) \cdot \cos x\right) \cdot \frac{-3}{2}\right)}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y, \frac{-3}{2}, \left(\left(1 - \sqrt{5}\right) \cdot \cos x\right) \cdot \frac{-3}{2}\right)}} \]

      11. lower-*.f64 62.3

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y, -1.5, \color{blue}{\left(\left(1 - \sqrt{5}\right) \cdot \cos x\right) \cdot -1.5}\right)} \]

   8. Applied rewrites 62.3%

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y, -1.5, \left(\left(1 - \sqrt{5}\right) \cdot \cos x\right) \cdot -1.5\right)}} \]

   if -3.7000000000000002e-6 < x < 1.11999999999999995e-5

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-cos.f64 62.1

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 62.1%

      \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right)}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(\color{blue}{3} - \sqrt{5}\right), \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right), \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      9. lower-sqrt.f64 59.4

         \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   9. Applied rewrites 59.4%

      \[\leadsto \frac{2 + \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   if 1.11999999999999995e-5 < 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

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 62.3

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 79.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 3\\ t_2 := t\_1 \cdot \cos y\\ t_3 := 1 - \sqrt{5}\\ t_4 := t\_3 \cdot \cos x\\ \mathbf{if}\;x \leq -0.0028:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{3 + \mathsf{fma}\left(t\_2, -1.5, t\_4 \cdot -1.5\right)}\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos y, t\_4\right), -1.5, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-0.0625 \cdot \left(t\_0 \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_3, \cos x, t\_2\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- (sqrt 5.0) 3.0))
        (t_2 (* t_1 (cos y)))
        (t_3 (- 1.0 (sqrt 5.0)))
        (t_4 (* t_3 (cos x))))
   (if (<= x -0.0028)
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_0))))
      (+ 3.0 (fma t_2 -1.5 (* t_4 -1.5))))
     (if (<= x 0.001)
       (/
        (-
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- (cos y) 1.0))))
         2.0)
        (- (fma (fma t_1 (cos y) t_4) -1.5 3.0)))
       (/
        (*
         (-
          (* (* -0.0625 (* t_0 (sqrt 2.0))) (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          -2.0)
         0.3333333333333333)
        (fma -0.5 (fma t_3 (cos x) t_2) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 3.0;
	double t_2 = t_1 * cos(y);
	double t_3 = 1.0 - sqrt(5.0);
	double t_4 = t_3 * cos(x);
	double tmp;
	if (x <= -0.0028) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_0)))) / (3.0 + fma(t_2, -1.5, (t_4 * -1.5)));
	} else if (x <= 0.001) {
		tmp = ((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / -fma(fma(t_1, cos(y), t_4), -1.5, 3.0);
	} else {
		tmp = ((((-0.0625 * (t_0 * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - -2.0) * 0.3333333333333333) / fma(-0.5, fma(t_3, cos(x), t_2), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 3.0)
	t_2 = Float64(t_1 * cos(y))
	t_3 = Float64(1.0 - sqrt(5.0))
	t_4 = Float64(t_3 * cos(x))
	tmp = 0.0
	if (x <= -0.0028)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_0)))) / Float64(3.0 + fma(t_2, -1.5, Float64(t_4 * -1.5))));
	elseif (x <= 0.001)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(-fma(fma(t_1, cos(y), t_4), -1.5, 3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.0625 * Float64(t_0 * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) - -2.0) * 0.3333333333333333) / fma(-0.5, fma(t_3, cos(x), t_2), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0028], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(t$95$2 * -1.5 + N[(t$95$4 * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.001], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / (-N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$4), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision], N[(N[(N[(N[(N[(-0.0625 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00279999999999999997

   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

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 62.3

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. Applied rewrites 62.3%

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \left(-0.5 \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)\right) \cdot 3}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)\right) \cdot 3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{3 \cdot \left(\frac{-1}{2} \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 3 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\left(3 \cdot \frac{-1}{2}\right) \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\frac{-3}{2}} \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{-3}{2} \cdot \color{blue}{\left(\left(1 - \sqrt{5}\right) \cdot \cos x + \left(\sqrt{5} - 3\right) \cdot \cos y\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{-3}{2} \cdot \left(\color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x} + \left(\sqrt{5} - 3\right) \cdot \cos y\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{-3}{2} \cdot \color{blue}{\left(\left(\sqrt{5} - 3\right) \cdot \cos y + \left(1 - \sqrt{5}\right) \cdot \cos x\right)}} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\left(\left(\left(\sqrt{5} - 3\right) \cdot \cos y\right) \cdot \frac{-3}{2} + \left(\left(1 - \sqrt{5}\right) \cdot \cos x\right) \cdot \frac{-3}{2}\right)}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y, \frac{-3}{2}, \left(\left(1 - \sqrt{5}\right) \cdot \cos x\right) \cdot \frac{-3}{2}\right)}} \]

      11. lower-*.f64 62.3

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y, -1.5, \color{blue}{\left(\left(1 - \sqrt{5}\right) \cdot \cos x\right) \cdot -1.5}\right)} \]

   8. Applied rewrites 62.3%

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y, -1.5, \left(\left(1 - \sqrt{5}\right) \cdot \cos x\right) \cdot -1.5\right)}} \]

   if -0.00279999999999999997 < x < 1e-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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x + \left(\sqrt{5} - 3\right) \cdot \cos y}, \frac{-3}{2}, 3\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 3\right) \cdot \cos y + \left(1 - \sqrt{5}\right) \cdot \cos x}, \frac{-3}{2}, 3\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 3\right) \cdot \cos y} + \left(1 - \sqrt{5}\right) \cdot \cos x, \frac{-3}{2}, 3\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right)}, \frac{-3}{2}, 3\right)} \]

      5. lower-*.f64 99.3

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x}\right), -1.5, 3\right)} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right)}, -1.5, 3\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - \color{blue}{2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      9. lower-cos.f64 62.1

         \[\leadsto \frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   10. Applied rewrites 62.1%

       \[\leadsto \frac{\color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   if 1e-3 < 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

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 62.3

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 79.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 3\\ t_1 := 1 - \sqrt{5}\\ t_2 := -\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos y, t\_1 \cdot \cos x\right), -1.5, 3\right)\\ \mathbf{if}\;x \leq -0.0028:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{t\_2}\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_1, \cos x, t\_0 \cdot \cos y\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 3.0))
        (t_1 (- 1.0 (sqrt 5.0)))
        (t_2 (- (fma (fma t_0 (cos y) (* t_1 (cos x))) -1.5 3.0))))
   (if (<= x -0.0028)
     (/
      (- (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- 1.0 (cos x))))) 2.0)
      t_2)
     (if (<= x 0.001)
       (/
        (-
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- (cos y) 1.0))))
         2.0)
        t_2)
       (/
        (*
         (-
          (*
           (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
           (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          -2.0)
         0.3333333333333333)
        (fma -0.5 (fma t_1 (cos x) (* t_0 (cos y))) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 3.0;
	double t_1 = 1.0 - sqrt(5.0);
	double t_2 = -fma(fma(t_0, cos(y), (t_1 * cos(x))), -1.5, 3.0);
	double tmp;
	if (x <= -0.0028) {
		tmp = ((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / t_2;
	} else if (x <= 0.001) {
		tmp = ((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / t_2;
	} else {
		tmp = ((((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - -2.0) * 0.3333333333333333) / fma(-0.5, fma(t_1, cos(x), (t_0 * cos(y))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 3.0)
	t_1 = Float64(1.0 - sqrt(5.0))
	t_2 = Float64(-fma(fma(t_0, cos(y), Float64(t_1 * cos(x))), -1.5, 3.0))
	tmp = 0.0
	if (x <= -0.0028)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / t_2);
	elseif (x <= 0.001)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) - -2.0) * 0.3333333333333333) / fma(-0.5, fma(t_1, cos(x), Float64(t_0 * cos(y))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[(N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])}, If[LessEqual[x, -0.0028], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 0.001], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00279999999999999997

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x + \left(\sqrt{5} - 3\right) \cdot \cos y}, \frac{-3}{2}, 3\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 3\right) \cdot \cos y + \left(1 - \sqrt{5}\right) \cdot \cos x}, \frac{-3}{2}, 3\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 3\right) \cdot \cos y} + \left(1 - \sqrt{5}\right) \cdot \cos x, \frac{-3}{2}, 3\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right)}, \frac{-3}{2}, 3\right)} \]

      5. lower-*.f64 99.3

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x}\right), -1.5, 3\right)} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right)}, -1.5, 3\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - \color{blue}{2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      9. lower-cos.f64 62.3

         \[\leadsto \frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   10. Applied rewrites 62.3%

       \[\leadsto \frac{\color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   if -0.00279999999999999997 < x < 1e-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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x + \left(\sqrt{5} - 3\right) \cdot \cos y}, \frac{-3}{2}, 3\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 3\right) \cdot \cos y + \left(1 - \sqrt{5}\right) \cdot \cos x}, \frac{-3}{2}, 3\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 3\right) \cdot \cos y} + \left(1 - \sqrt{5}\right) \cdot \cos x, \frac{-3}{2}, 3\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right)}, \frac{-3}{2}, 3\right)} \]

      5. lower-*.f64 99.3

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x}\right), -1.5, 3\right)} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right)}, -1.5, 3\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - \color{blue}{2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      9. lower-cos.f64 62.1

         \[\leadsto \frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   10. Applied rewrites 62.1%

       \[\leadsto \frac{\color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   if 1e-3 < 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

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 62.3

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 79.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 3\\ t_1 := 1 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(t\_1, \cos x, t\_0 \cdot \cos y\right)\\ \mathbf{if}\;x \leq -0.0028:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos y, t\_1 \cdot \cos x\right), -1.5, 3\right)}\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(t\_2, -1.5, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, t\_2, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 3.0))
        (t_1 (- 1.0 (sqrt 5.0)))
        (t_2 (fma t_1 (cos x) (* t_0 (cos y)))))
   (if (<= x -0.0028)
     (/
      (- (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- 1.0 (cos x))))) 2.0)
      (- (fma (fma t_0 (cos y) (* t_1 (cos x))) -1.5 3.0)))
     (if (<= x 0.001)
       (/
        (-
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- (cos y) 1.0))))
         2.0)
        (- (fma t_2 -1.5 3.0)))
       (/
        (*
         (-
          (*
           (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
           (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          -2.0)
         0.3333333333333333)
        (fma -0.5 t_2 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 3.0;
	double t_1 = 1.0 - sqrt(5.0);
	double t_2 = fma(t_1, cos(x), (t_0 * cos(y)));
	double tmp;
	if (x <= -0.0028) {
		tmp = ((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / -fma(fma(t_0, cos(y), (t_1 * cos(x))), -1.5, 3.0);
	} else if (x <= 0.001) {
		tmp = ((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / -fma(t_2, -1.5, 3.0);
	} else {
		tmp = ((((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - -2.0) * 0.3333333333333333) / fma(-0.5, t_2, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 3.0)
	t_1 = Float64(1.0 - sqrt(5.0))
	t_2 = fma(t_1, cos(x), Float64(t_0 * cos(y)))
	tmp = 0.0
	if (x <= -0.0028)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(-fma(fma(t_0, cos(y), Float64(t_1 * cos(x))), -1.5, 3.0)));
	elseif (x <= 0.001)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(-fma(t_2, -1.5, 3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) - -2.0) * 0.3333333333333333) / fma(-0.5, t_2, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0028], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / (-N[(N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 0.001], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / (-N[(t$95$2 * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision], N[(N[(N[(N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00279999999999999997

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x + \left(\sqrt{5} - 3\right) \cdot \cos y}, \frac{-3}{2}, 3\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 3\right) \cdot \cos y + \left(1 - \sqrt{5}\right) \cdot \cos x}, \frac{-3}{2}, 3\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 3\right) \cdot \cos y} + \left(1 - \sqrt{5}\right) \cdot \cos x, \frac{-3}{2}, 3\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right)}, \frac{-3}{2}, 3\right)} \]

      5. lower-*.f64 99.3

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x}\right), -1.5, 3\right)} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right)}, -1.5, 3\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - \color{blue}{2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      9. lower-cos.f64 62.3

         \[\leadsto \frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   10. Applied rewrites 62.3%

       \[\leadsto \frac{\color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   if -0.00279999999999999997 < x < 1e-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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), \frac{-3}{2}, 3\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - \color{blue}{2}}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), \frac{-3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), \frac{-3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), \frac{-3}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), \frac{-3}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), \frac{-3}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), \frac{-3}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), \frac{-3}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), \frac{-3}{2}, 3\right)} \]

      9. lower-cos.f64 62.1

         \[\leadsto \frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)} \]

   if 1e-3 < 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

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 62.3

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 3\\ t_1 := 1 - \sqrt{5}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos y, t\_1 \cdot \cos x\right), -1.5, 3\right)}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + -1.5 \cdot \left(\left(1 + \cos y \cdot t\_0\right) - \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_1, \cos x, t\_0 \cdot \cos y\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 3.0)) (t_1 (- 1.0 (sqrt 5.0))))
   (if (<= x -2.7e-6)
     (/
      (- (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- 1.0 (cos x))))) 2.0)
      (- (fma (fma t_0 (cos y) (* t_1 (cos x))) -1.5 3.0)))
     (if (<= x 2.7e-6)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* -1.5 (- (+ 1.0 (* (cos y) t_0)) (sqrt 5.0)))))
       (/
        (*
         (-
          (*
           (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
           (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          -2.0)
         0.3333333333333333)
        (fma -0.5 (fma t_1 (cos x) (* t_0 (cos y))) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 3.0;
	double t_1 = 1.0 - sqrt(5.0);
	double tmp;
	if (x <= -2.7e-6) {
		tmp = ((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / -fma(fma(t_0, cos(y), (t_1 * cos(x))), -1.5, 3.0);
	} else if (x <= 2.7e-6) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (-1.5 * ((1.0 + (cos(y) * t_0)) - sqrt(5.0))));
	} else {
		tmp = ((((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - -2.0) * 0.3333333333333333) / fma(-0.5, fma(t_1, cos(x), (t_0 * cos(y))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 3.0)
	t_1 = Float64(1.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -2.7e-6)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(-fma(fma(t_0, cos(y), Float64(t_1 * cos(x))), -1.5, 3.0)));
	elseif (x <= 2.7e-6)
		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(1.0 + Float64(cos(y) * t_0)) - sqrt(5.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) - -2.0) * 0.3333333333333333) / fma(-0.5, fma(t_1, cos(x), Float64(t_0 * cos(y))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-6], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / (-N[(N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + 3.0), $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 2.7e-6], 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[(1.0 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.69999999999999998e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), -1.5, 3\right)}} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x + \left(\sqrt{5} - 3\right) \cdot \cos y}, \frac{-3}{2}, 3\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 3\right) \cdot \cos y + \left(1 - \sqrt{5}\right) \cdot \cos x}, \frac{-3}{2}, 3\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 3\right) \cdot \cos y} + \left(1 - \sqrt{5}\right) \cdot \cos x, \frac{-3}{2}, 3\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right)}, \frac{-3}{2}, 3\right)} \]

      5. lower-*.f64 99.3

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \color{blue}{\left(1 - \sqrt{5}\right) \cdot \cos x}\right), -1.5, 3\right)} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right)}, -1.5, 3\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - \color{blue}{2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), \frac{-3}{2}, 3\right)} \]

      9. lower-cos.f64 62.3

         \[\leadsto \frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   10. Applied rewrites 62.3%

       \[\leadsto \frac{\color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{-\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 3, \cos y, \left(1 - \sqrt{5}\right) \cdot \cos x\right), -1.5, 3\right)} \]

   if -2.69999999999999998e-6 < x < 2.69999999999999998e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{-3}{2} \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{-3}{2} \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]

   6. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + -1.5 \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]

   if 2.69999999999999998e-6 < 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

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 62.3

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\\ t_1 := \sqrt{5} - 3\\ t_2 := \mathsf{fma}\left(1 - \sqrt{5}, \cos x, t\_1 \cdot \cos y\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(-1.5, t\_2, 3\right)}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + -1.5 \cdot \left(\left(1 + \cos y \cdot t\_1\right) - \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, t\_2, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (-
          (*
           (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
           (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          -2.0))
        (t_1 (- (sqrt 5.0) 3.0))
        (t_2 (fma (- 1.0 (sqrt 5.0)) (cos x) (* t_1 (cos y)))))
   (if (<= x -2.7e-6)
     (/ t_0 (fma -1.5 t_2 3.0))
     (if (<= x 2.7e-6)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* -1.5 (- (+ 1.0 (* (cos y) t_1)) (sqrt 5.0)))))
       (/ (* t_0 0.3333333333333333) (fma -0.5 t_2 1.0))))))

C

double code(double x, double y) {
	double t_0 = ((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - -2.0;
	double t_1 = sqrt(5.0) - 3.0;
	double t_2 = fma((1.0 - sqrt(5.0)), cos(x), (t_1 * cos(y)));
	double tmp;
	if (x <= -2.7e-6) {
		tmp = t_0 / fma(-1.5, t_2, 3.0);
	} else if (x <= 2.7e-6) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (-1.5 * ((1.0 + (cos(y) * t_1)) - sqrt(5.0))));
	} else {
		tmp = (t_0 * 0.3333333333333333) / fma(-0.5, t_2, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) - -2.0)
	t_1 = Float64(sqrt(5.0) - 3.0)
	t_2 = fma(Float64(1.0 - sqrt(5.0)), cos(x), Float64(t_1 * cos(y)))
	tmp = 0.0
	if (x <= -2.7e-6)
		tmp = Float64(t_0 / fma(-1.5, t_2, 3.0));
	elseif (x <= 2.7e-6)
		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(1.0 + Float64(cos(y) * t_1)) - sqrt(5.0)))));
	else
		tmp = Float64(Float64(t_0 * 0.3333333333333333) / fma(-0.5, t_2, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-6], N[(t$95$0 / N[(-1.5 * t$95$2 + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-6], 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[(1.0 + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.69999999999999998e-6

   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

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 62.3

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. Applied rewrites 62.3%

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \left(-0.5 \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)\right) \cdot 3}} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2}{\mathsf{fma}\left(-1.5, \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), 3\right)}} \]

   if -2.69999999999999998e-6 < x < 2.69999999999999998e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{-3}{2} \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{-3}{2} \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]

   6. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + -1.5 \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]

   if 2.69999999999999998e-6 < 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

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 62.3

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 3\\ t_1 := \frac{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2}{\mathsf{fma}\left(-1.5, \mathsf{fma}\left(1 - \sqrt{5}, \cos x, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + -1.5 \cdot \left(\left(1 + \cos y \cdot t\_0\right) - \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 3.0))
        (t_1
         (/
          (-
           (*
            (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
            (- 0.5 (* 0.5 (cos (* 2.0 x)))))
           -2.0)
          (fma -1.5 (fma (- 1.0 (sqrt 5.0)) (cos x) (* t_0 (cos y))) 3.0))))
   (if (<= x -2.7e-6)
     t_1
     (if (<= x 2.7e-6)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* -1.5 (- (+ 1.0 (* (cos y) t_0)) (sqrt 5.0)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 3.0;
	double t_1 = (((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - -2.0) / fma(-1.5, fma((1.0 - sqrt(5.0)), cos(x), (t_0 * cos(y))), 3.0);
	double tmp;
	if (x <= -2.7e-6) {
		tmp = t_1;
	} else if (x <= 2.7e-6) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (-1.5 * ((1.0 + (cos(y) * t_0)) - sqrt(5.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 3.0)
	t_1 = Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) - -2.0) / fma(-1.5, fma(Float64(1.0 - sqrt(5.0)), cos(x), Float64(t_0 * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -2.7e-6)
		tmp = t_1;
	elseif (x <= 2.7e-6)
		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(1.0 + Float64(cos(y) * t_0)) - sqrt(5.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / N[(-1.5 * N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-6], t$95$1, If[LessEqual[x, 2.7e-6], 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[(1.0 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999998e-6 or 2.69999999999999998e-6 < 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

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 62.3

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. Applied rewrites 62.3%

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \left(-0.5 \cdot \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right)\right) \cdot 3}} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2}{\mathsf{fma}\left(-1.5, \mathsf{fma}\left(1 - \sqrt{5}, \cos x, \left(\sqrt{5} - 3\right) \cdot \cos y\right), 3\right)}} \]

   if -2.69999999999999998e-6 < x < 2.69999999999999998e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{-3}{2} \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{-3}{2} \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]

   6. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + -1.5 \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{3 + -1.5 \cdot \left(\left(\sqrt{5} + \cos x \cdot \left(1 - \sqrt{5}\right)\right) - 3\right)}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + -1.5 \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(t\_0 \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0)))
   (if (<= x -3.7e-6)
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_0))))
      (+ 3.0 (* -1.5 (- (+ (sqrt 5.0) (* (cos x) (- 1.0 (sqrt 5.0)))) 3.0))))
     (if (<= x 1.15e-5)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* -1.5 (- (+ 1.0 (* (cos y) (- (sqrt 5.0) 3.0))) (sqrt 5.0)))))
       (*
        (/
         (fma
          (* -0.0625 (* t_0 (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x))))
          2.0)
         (fma (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 0.5 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double tmp;
	if (x <= -3.7e-6) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_0)))) / (3.0 + (-1.5 * ((sqrt(5.0) + (cos(x) * (1.0 - sqrt(5.0)))) - 3.0)));
	} else if (x <= 1.15e-5) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (-1.5 * ((1.0 + (cos(y) * (sqrt(5.0) - 3.0))) - sqrt(5.0))));
	} else {
		tmp = (fma((-0.0625 * (t_0 * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 0.5, 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	tmp = 0.0
	if (x <= -3.7e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_0)))) / Float64(3.0 + Float64(-1.5 * Float64(Float64(sqrt(5.0) + Float64(cos(x) * Float64(1.0 - sqrt(5.0)))) - 3.0))));
	elseif (x <= 1.15e-5)
		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(1.0 + Float64(cos(y) * Float64(sqrt(5.0) - 3.0))) - sqrt(5.0)))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(t_0 * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 0.5, 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -3.7e-6], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(-1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e-5], 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[(1.0 + N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7000000000000002e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos x \cdot \left(1 - \sqrt{5}\right)\right) - 3\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos x \cdot \left(1 - \sqrt{5}\right)\right) - 3\right)}} \]

   6. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + -1.5 \cdot \left(\left(\sqrt{5} + \cos x \cdot \left(1 - \sqrt{5}\right)\right) - 3\right)}} \]

   if -3.7000000000000002e-6 < x < 1.15e-5

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \frac{\mathsf{fma}\left(\cos y, \sqrt{5} - 3, \cos x \cdot \left(1 - \sqrt{5}\right)\right)}{-2} \cdot 3}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{-3}{2} \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{-3}{2} \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]

   6. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + -1.5 \cdot \left(\left(1 + \cos y \cdot \left(\sqrt{5} - 3\right)\right) - \sqrt{5}\right)}} \]

   if 1.15e-5 < 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 60.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 60.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
    (- 0.5 (* 0.5 (cos (* 2.0 x))))
    2.0)
   (fma (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 0.5 1.0))
  0.3333333333333333))

C

double code(double x, double y) {
	return (fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 0.5, 1.0)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 0.5, 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 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 60.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

6. Applied rewrites 60.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]
7. Add Preprocessing

Alternative 30: 42.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \end{array} \]

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 60.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \cdot \frac{2}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 42.8%

   \[\leadsto 0.3333333333333333 \cdot \frac{2}{\color{blue}{1} + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
8. Add Preprocessing

Alternative 31: 40.2% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(1 - \sqrt{5}\right), 0.5, 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (fma (- (- 3.0 (sqrt 5.0)) (- 1.0 (sqrt 5.0))) 0.5 1.0)))

C

double code(double x, double y) {
	return 0.6666666666666666 / fma(((3.0 - sqrt(5.0)) - (1.0 - sqrt(5.0))), 0.5, 1.0);
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 / fma(Float64(Float64(3.0 - sqrt(5.0)) - Float64(1.0 - sqrt(5.0))), 0.5, 1.0))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] - N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 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 60.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

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 40.2

      \[\leadsto \frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, 3 - \sqrt{5}, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)} \]

7. Applied rewrites 40.2%

   \[\leadsto \frac{0.6666666666666666}{\color{blue}{1 + \mathsf{fma}\left(0.5, 3 - \sqrt{5}, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{3 - \sqrt{5}}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1} \]

   3. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1} \]

   5. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 1} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + 1} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), \frac{1}{2}, 1\right)} \]

   8. add-flip N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(\mathsf{neg}\left(\left(\sqrt{5} - 1\right)\right)\right), \frac{1}{2}, 1\right)} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(\mathsf{neg}\left(\left(\sqrt{5} - 1\right)\right)\right), \frac{1}{2}, 1\right)} \]

   10. sub-negate-rev N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(1 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

   11. lift--.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(1 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

   12. lower--.f64 40.2

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(1 - \sqrt{5}\right), 0.5, 1\right)} \]

9. Applied rewrites 40.2%

   \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(1 - \sqrt{5}\right), 0.5, 1\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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))))))