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

Percentage Accurate: 99.3% → 99.4%

Time: 16.4s

Alternatives: 25

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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (- (cos x) (cos y)) (sqrt 2.0))
   (* (fma -0.0625 (sin y) (sin x)) (fma -0.0625 (sin x) (sin y)))
   2.0)
  (fma
   1.5
   (fma (cos x) (- (sqrt 5.0) 1.0) (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
   3.0)))

C

double code(double x, double y) {
	return fma(((cos(x) - cos(y)) * sqrt(2.0)), (fma(-0.0625, sin(y), sin(x)) * fma(-0.0625, sin(x), sin(y))), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(fma(-0.0625, sin(y), sin(x)) * fma(-0.0625, sin(x), sin(y))), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

8. Taylor expanded in x around inf

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

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]
11. Step-by-step derivation

12. Applied rewrites 99.4%

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

Alternative 2: 61.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)} \leq 0.548:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos y, t\_1\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos y, t\_1 \cdot \cos x\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (- (sqrt 5.0) 1.0)))
   (if (<=
        (/
         (+
          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 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))
        0.548)
     (/
      (fma
       -0.020833333333333332
       (* (* (- 1.0 (cos y)) (sqrt 2.0)) (pow (sin y) 2.0))
       0.6666666666666666)
      (fma (fma t_0 (cos y) t_1) 0.5 1.0))
     (/ 2.0 (fma 1.5 (fma t_0 (cos y) (* t_1 (cos x))) 3.0)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if (((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 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))))) <= 0.548) {
		tmp = fma(-0.020833333333333332, (((1.0 - cos(y)) * sqrt(2.0)) * pow(sin(y), 2.0)), 0.6666666666666666) / fma(fma(t_0, cos(y), t_1), 0.5, 1.0);
	} else {
		tmp = 2.0 / fma(1.5, fma(t_0, cos(y), (t_1 * cos(x))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (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(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y))))) <= 0.548)
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * (sin(y) ^ 2.0)), 0.6666666666666666) / fma(fma(t_0, cos(y), t_1), 0.5, 1.0));
	else
		tmp = Float64(2.0 / fma(1.5, fma(t_0, cos(y), Float64(t_1 * cos(x))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[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[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.548], N[(N[(-0.020833333333333332 * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(1.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) #s(literal 16 binary64)))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) #s(literal 16 binary64)))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 #s(literal 3 binary64) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (-.f64 (sqrt.f64 #s(literal 5 binary64)) #s(literal 1 binary64)) #s(literal 2 binary64)) (cos.f64 x))) (*.f64 (/.f64 (-.f64 #s(literal 3 binary64) (sqrt.f64 #s(literal 5 binary64))) #s(literal 2 binary64)) (cos.f64 y))))) < 0.54800000000000004

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 67.2

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

   5. Applied rewrites 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 67.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 73.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 0.5, 1\right)}} \]

   if 0.54800000000000004 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) #s(literal 16 binary64)))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) #s(literal 16 binary64)))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 #s(literal 3 binary64) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (-.f64 (sqrt.f64 #s(literal 5 binary64)) #s(literal 1 binary64)) #s(literal 2 binary64)) (cos.f64 x))) (*.f64 (/.f64 (-.f64 #s(literal 3 binary64) (sqrt.f64 #s(literal 5 binary64))) #s(literal 2 binary64)) (cos.f64 y)))))

   1. Initial program 98.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 58.8

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

   8. Applied rewrites 58.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 23.8%

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

   12. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. associate-*r* N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

   14. Applied rewrites 26.2%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

6. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

8. Taylor expanded in x around inf

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

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

12. Applied rewrites 99.4%

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

Alternative 5: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.038 \lor \neg \left(x \leq 0.028\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \left(\cos x - \cos y\right), \sqrt{2} \cdot \sin x, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t\_2\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right)}{\mathsf{fma}\left(t\_1 \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos y, t\_1\right), 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (sin y) (/ (sin x) 16.0))))
   (if (or (<= x -0.038) (not (<= x 0.028)))
     (/
      (fma (* t_2 (- (cos x) (cos y))) (* (sqrt 2.0) (sin x)) 2.0)
      (fma 1.5 (fma (cos x) t_1 (* t_0 (cos y))) 3.0))
     (/
      (+
       2.0
       (*
        (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_2)
        (fma (* x x) -0.5 (- 1.0 (cos y)))))
      (fma
       (* t_1 (fma 0.0625 (* x x) -0.75))
       (* x x)
       (fma 1.5 (fma t_0 (cos y) t_1) 3.0))))))

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 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if ((x <= -0.038) || !(x <= 0.028)) {
		tmp = fma((t_2 * (cos(x) - cos(y))), (sqrt(2.0) * sin(x)), 2.0) / fma(1.5, fma(cos(x), t_1, (t_0 * cos(y))), 3.0);
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_2) * fma((x * x), -0.5, (1.0 - cos(y))))) / fma((t_1 * fma(0.0625, (x * x), -0.75)), (x * x), fma(1.5, fma(t_0, cos(y), t_1), 3.0));
	}
	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 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if ((x <= -0.038) || !(x <= 0.028))
		tmp = Float64(fma(Float64(t_2 * Float64(cos(x) - cos(y))), Float64(sqrt(2.0) * sin(x)), 2.0) / fma(1.5, fma(cos(x), t_1, Float64(t_0 * cos(y))), 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_2) * fma(Float64(x * x), -0.5, Float64(1.0 - cos(y))))) / fma(Float64(t_1 * fma(0.0625, Float64(x * x), -0.75)), Float64(x * x), fma(1.5, fma(t_0, cos(y), t_1), 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.038], N[Not[LessEqual[x, 0.028]], $MachinePrecision]], N[(N[(N[(t$95$2 * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], 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] * t$95$2), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * N[(0.0625 * N[(x * x), $MachinePrecision] + -0.75), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(1.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0379999999999999991 or 0.0280000000000000006 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 66.7

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 66.7%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 66.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), \sqrt{2} \cdot \sin x, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if -0.0379999999999999991 < x < 0.0280000000000000006

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right), {x}^{2}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      2. associate--l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{2}} + \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1 - \cos y\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1 - \cos y\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1 - \cos y\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, \color{blue}{1 - \cos y}\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      8. lower-cos.f64 99.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.038 \lor \neg \left(x \leq 0.028\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), \sqrt{2} \cdot \sin x, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \sqrt{5} - 1\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := 3 - \sqrt{5}\\ t_4 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, t\_3 \cdot \cos y\right), 3\right)\\ t_5 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.038:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot t\_5, t\_0, 2\right)}{t\_4}\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t\_2\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right)}{\mathsf{fma}\left(t\_1 \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_3, \cos y, t\_1\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_0 \cdot t\_5, 2\right)}{t\_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (sin x)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (sin y) (/ (sin x) 16.0)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (fma 1.5 (fma (cos x) t_1 (* t_3 (cos y))) 3.0))
        (t_5 (- (cos x) (cos y))))
   (if (<= x -0.038)
     (/ (fma (* t_2 t_5) t_0 2.0) t_4)
     (if (<= x 0.028)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_2)
          (fma (* x x) -0.5 (- 1.0 (cos y)))))
        (fma
         (* t_1 (fma 0.0625 (* x x) -0.75))
         (* x x)
         (fma 1.5 (fma t_3 (cos y) t_1) 3.0)))
       (/ (fma t_2 (* t_0 t_5) 2.0) t_4)))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * sin(x);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = sin(y) - (sin(x) / 16.0);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = fma(1.5, fma(cos(x), t_1, (t_3 * cos(y))), 3.0);
	double t_5 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.038) {
		tmp = fma((t_2 * t_5), t_0, 2.0) / t_4;
	} else if (x <= 0.028) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_2) * fma((x * x), -0.5, (1.0 - cos(y))))) / fma((t_1 * fma(0.0625, (x * x), -0.75)), (x * x), fma(1.5, fma(t_3, cos(y), t_1), 3.0));
	} else {
		tmp = fma(t_2, (t_0 * t_5), 2.0) / t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * sin(x))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = fma(1.5, fma(cos(x), t_1, Float64(t_3 * cos(y))), 3.0)
	t_5 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.038)
		tmp = Float64(fma(Float64(t_2 * t_5), t_0, 2.0) / t_4);
	elseif (x <= 0.028)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_2) * fma(Float64(x * x), -0.5, Float64(1.0 - cos(y))))) / fma(Float64(t_1 * fma(0.0625, Float64(x * x), -0.75)), Float64(x * x), fma(1.5, fma(t_3, cos(y), t_1), 3.0)));
	else
		tmp = Float64(fma(t_2, Float64(t_0 * t_5), 2.0) / t_4);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.038], N[(N[(N[(t$95$2 * t$95$5), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[x, 0.028], 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] * t$95$2), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * N[(0.0625 * N[(x * x), $MachinePrecision] + -0.75), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(1.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(t$95$0 * t$95$5), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$4), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0379999999999999991

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 66.6

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 66.6%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 66.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), \sqrt{2} \cdot \sin x, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if -0.0379999999999999991 < x < 0.0280000000000000006

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right), {x}^{2}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      2. associate--l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{2}} + \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1 - \cos y\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1 - \cos y\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1 - \cos y\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, \color{blue}{1 - \cos y}\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      8. lower-cos.f64 99.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   if 0.0280000000000000006 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 66.8

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 66.8%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

   10. Applied rewrites 66.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin x}^{2}\\ t_2 := 3 - \sqrt{5}\\ t_3 := t\_2 \cdot \cos y\\ \mathbf{if}\;x \leq -0.235:\\ \;\;\;\;\frac{2 + \left(\left(t\_1 \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_3\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.031:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right)}{\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot t\_1, 0.6666666666666666\right)}{\frac{\mathsf{fma}\left(t\_0, \cos x, t\_3\right)}{2} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (* t_2 (cos y))))
   (if (<= x -0.235)
     (/
      (+ 2.0 (* (* (* t_1 -0.0625) (sqrt 2.0)) (- (cos x) (cos y))))
      (fma 1.5 (fma (cos x) t_0 t_3) 3.0))
     (if (<= x 0.031)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (/ (sin x) 16.0)))
          (fma (* x x) -0.5 (- 1.0 (cos y)))))
        (fma
         (* t_0 (fma 0.0625 (* x x) -0.75))
         (* x x)
         (fma 1.5 (fma t_2 (cos y) t_0) 3.0)))
       (/
        (fma
         -0.020833333333333332
         (* (* (- (cos x) 1.0) (sqrt 2.0)) t_1)
         0.6666666666666666)
        (+ (/ (fma t_0 (cos x) t_3) 2.0) 1.0))))))

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 = 3.0 - sqrt(5.0);
	double t_3 = t_2 * cos(y);
	double tmp;
	if (x <= -0.235) {
		tmp = (2.0 + (((t_1 * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y)))) / fma(1.5, fma(cos(x), t_0, t_3), 3.0);
	} else if (x <= 0.031) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * fma((x * x), -0.5, (1.0 - cos(y))))) / fma((t_0 * fma(0.0625, (x * x), -0.75)), (x * x), fma(1.5, fma(t_2, cos(y), t_0), 3.0));
	} else {
		tmp = fma(-0.020833333333333332, (((cos(x) - 1.0) * sqrt(2.0)) * t_1), 0.6666666666666666) / ((fma(t_0, cos(x), t_3) / 2.0) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(t_2 * cos(y))
	tmp = 0.0
	if (x <= -0.235)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(t_1 * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / fma(1.5, fma(cos(x), t_0, t_3), 3.0));
	elseif (x <= 0.031)
		tmp = 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))) * fma(Float64(x * x), -0.5, Float64(1.0 - cos(y))))) / fma(Float64(t_0 * fma(0.0625, Float64(x * x), -0.75)), Float64(x * x), fma(1.5, fma(t_2, cos(y), t_0), 3.0)));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * t_1), 0.6666666666666666) / Float64(Float64(fma(t_0, cos(x), t_3) / 2.0) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.235], N[(N[(2.0 + N[(N[(N[(t$95$1 * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.031], 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[(x * x), $MachinePrecision] * -0.5 + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[(0.0625 * N[(x * x), $MachinePrecision] + -0.75), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(1.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.23499999999999999

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 63.4

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

   8. Applied rewrites 63.4%

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

   if -0.23499999999999999 < x < 0.031

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right), {x}^{2}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      2. associate--l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{2}} + \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1 - \cos y\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1 - \cos y\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{2}, 1 - \cos y\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, \color{blue}{1 - \cos y}\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      8. lower-cos.f64 99.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   if 0.031 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 63.4

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

   5. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 63.5%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{48}} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{48}\right), {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{48}}, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \color{blue}{{\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      16. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\color{blue}{\sin x}}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      17. metadata-eval 63.7

          \[\leadsto \frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \color{blue}{0.6666666666666666}\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

   10. Applied rewrites 63.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, {\sin y}^{2} \cdot -0.0625, 2\right)\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.00032:\\ \;\;\;\;\frac{t\_2}{3 \cdot \mathsf{fma}\left(\frac{t\_3}{2}, \cos y, \mathsf{fma}\left(\frac{t\_1}{2}, \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot y, \left(t\_0 \cdot 1.00390625\right) \cdot \sin x, \mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, t\_0 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_3\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (fma
          (* (- (cos x) (cos y)) (sqrt 2.0))
          (* (pow (sin y) 2.0) -0.0625)
          2.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= y -0.00032)
     (/ t_2 (* 3.0 (fma (/ t_3 2.0) (cos y) (fma (/ t_1 2.0) (cos x) 1.0))))
     (if (<= y 1.72e-6)
       (/
        (*
         0.3333333333333333
         (fma
          (* (sqrt 2.0) y)
          (* (* t_0 1.00390625) (sin x))
          (fma (* -0.0625 (pow (sin x) 2.0)) (* t_0 (sqrt 2.0)) 2.0)))
        (fma (fma t_1 (cos x) t_3) 0.5 1.0))
       (/
        t_2
        (fma
         1.5
         (fma (cos x) t_1 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma(((cos(x) - cos(y)) * sqrt(2.0)), (pow(sin(y), 2.0) * -0.0625), 2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -0.00032) {
		tmp = t_2 / (3.0 * fma((t_3 / 2.0), cos(y), fma((t_1 / 2.0), cos(x), 1.0)));
	} else if (y <= 1.72e-6) {
		tmp = (0.3333333333333333 * fma((sqrt(2.0) * y), ((t_0 * 1.00390625) * sin(x)), fma((-0.0625 * pow(sin(x), 2.0)), (t_0 * sqrt(2.0)), 2.0))) / fma(fma(t_1, cos(x), t_3), 0.5, 1.0);
	} else {
		tmp = t_2 / fma(1.5, fma(cos(x), t_1, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64((sin(y) ^ 2.0) * -0.0625), 2.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -0.00032)
		tmp = Float64(t_2 / Float64(3.0 * fma(Float64(t_3 / 2.0), cos(y), fma(Float64(t_1 / 2.0), cos(x), 1.0))));
	elseif (y <= 1.72e-6)
		tmp = Float64(Float64(0.3333333333333333 * fma(Float64(sqrt(2.0) * y), Float64(Float64(t_0 * 1.00390625) * sin(x)), fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(t_0 * sqrt(2.0)), 2.0))) / fma(fma(t_1, cos(x), t_3), 0.5, 1.0));
	else
		tmp = Float64(t_2 / fma(1.5, fma(cos(x), t_1, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	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] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00032], N[(t$95$2 / N[(3.0 * N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.72e-6], N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * y), $MachinePrecision] * N[(N[(t$95$0 * 1.00390625), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.20000000000000026e-4

   1. Initial program 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   4. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.2

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 61.0

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

   9. Applied rewrites 61.0%

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

   if -3.20000000000000026e-4 < y < 1.72e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 99.6%

       \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot y, \left(\left(\cos x - 1\right) \cdot 1.00390625\right) \cdot \sin x, \mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]

   if 1.72e-6 < y

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Applied rewrites 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 58.8%

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

Alternative 9: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, {\sin y}^{2} \cdot -0.0625, 2\right)\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.00032:\\ \;\;\;\;\frac{t\_2}{3 \cdot \mathsf{fma}\left(\frac{t\_3}{2}, \cos y, \mathsf{fma}\left(\frac{t\_1}{2}, \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot y, \left(t\_0 \cdot 1.00390625\right) \cdot \sin x, \mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, t\_0 \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_1, t\_3\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (fma
          (* (- (cos x) (cos y)) (sqrt 2.0))
          (* (pow (sin y) 2.0) -0.0625)
          2.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= y -0.00032)
     (/ t_2 (* 3.0 (fma (/ t_3 2.0) (cos y) (fma (/ t_1 2.0) (cos x) 1.0))))
     (if (<= y 1.72e-6)
       (*
        0.3333333333333333
        (/
         (fma
          (* (sqrt 2.0) y)
          (* (* t_0 1.00390625) (sin x))
          (fma (* (pow (sin x) 2.0) -0.0625) (* t_0 (sqrt 2.0)) 2.0))
         (fma 0.5 (fma (cos x) t_1 t_3) 1.0)))
       (/
        t_2
        (fma
         1.5
         (fma (cos x) t_1 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma(((cos(x) - cos(y)) * sqrt(2.0)), (pow(sin(y), 2.0) * -0.0625), 2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -0.00032) {
		tmp = t_2 / (3.0 * fma((t_3 / 2.0), cos(y), fma((t_1 / 2.0), cos(x), 1.0)));
	} else if (y <= 1.72e-6) {
		tmp = 0.3333333333333333 * (fma((sqrt(2.0) * y), ((t_0 * 1.00390625) * sin(x)), fma((pow(sin(x), 2.0) * -0.0625), (t_0 * sqrt(2.0)), 2.0)) / fma(0.5, fma(cos(x), t_1, t_3), 1.0));
	} else {
		tmp = t_2 / fma(1.5, fma(cos(x), t_1, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64((sin(y) ^ 2.0) * -0.0625), 2.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -0.00032)
		tmp = Float64(t_2 / Float64(3.0 * fma(Float64(t_3 / 2.0), cos(y), fma(Float64(t_1 / 2.0), cos(x), 1.0))));
	elseif (y <= 1.72e-6)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * y), Float64(Float64(t_0 * 1.00390625) * sin(x)), fma(Float64((sin(x) ^ 2.0) * -0.0625), Float64(t_0 * sqrt(2.0)), 2.0)) / fma(0.5, fma(cos(x), t_1, t_3), 1.0)));
	else
		tmp = Float64(t_2 / fma(1.5, fma(cos(x), t_1, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	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] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00032], N[(t$95$2 / N[(3.0 * N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.72e-6], N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * y), $MachinePrecision] * N[(N[(t$95$0 * 1.00390625), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.20000000000000026e-4

   1. Initial program 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   4. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.2

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 61.0

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

   9. Applied rewrites 61.0%

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

   if -3.20000000000000026e-4 < y < 1.72e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot y, \left(\left(\cos x - 1\right) \cdot 1.00390625\right) \cdot \sin x, \mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)}} \]

   if 1.72e-6 < y

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Applied rewrites 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 58.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00032:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, {\sin y}^{2} \cdot -0.0625, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot y, \left(\left(\cos x - 1\right) \cdot 1.00390625\right) \cdot \sin x, \mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, {\sin y}^{2} \cdot -0.0625, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(t\_0 \cdot \sqrt{2}, {\sin y}^{2} \cdot -0.0625, 2\right)\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_2}{3 \cdot \mathsf{fma}\left(\frac{t\_3}{2}, \cos y, \mathsf{fma}\left(\frac{t\_1}{2}, \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, t\_3\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (fma (* t_0 (sqrt 2.0)) (* (pow (sin y) 2.0) -0.0625) 2.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= y -7e-5)
     (/ t_2 (* 3.0 (fma (/ t_3 2.0) (cos y) (fma (/ t_1 2.0) (cos x) 1.0))))
     (if (<= y 1.72e-6)
       (/
        (+ 2.0 (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_0))
        (fma 1.5 (fma (cos x) t_1 t_3) 3.0))
       (/
        t_2
        (fma
         1.5
         (fma (cos x) t_1 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma((t_0 * sqrt(2.0)), (pow(sin(y), 2.0) * -0.0625), 2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -7e-5) {
		tmp = t_2 / (3.0 * fma((t_3 / 2.0), cos(y), fma((t_1 / 2.0), cos(x), 1.0)));
	} else if (y <= 1.72e-6) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma(1.5, fma(cos(x), t_1, t_3), 3.0);
	} else {
		tmp = t_2 / fma(1.5, fma(cos(x), t_1, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(Float64(t_0 * sqrt(2.0)), Float64((sin(y) ^ 2.0) * -0.0625), 2.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -7e-5)
		tmp = Float64(t_2 / Float64(3.0 * fma(Float64(t_3 / 2.0), cos(y), fma(Float64(t_1 / 2.0), cos(x), 1.0))));
	elseif (y <= 1.72e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(1.5, fma(cos(x), t_1, t_3), 3.0));
	else
		tmp = Float64(t_2 / fma(1.5, fma(cos(x), t_1, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	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] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e-5], N[(t$95$2 / N[(3.0 * N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.72e-6], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.9999999999999994e-5

   1. Initial program 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   4. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.2

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 61.0

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

   9. Applied rewrites 61.0%

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

   if -6.9999999999999994e-5 < y < 1.72e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

   if 1.72e-6 < y

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Applied rewrites 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 58.8%

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

Alternative 11: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := {\sin y}^{2} \cdot -0.0625\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(t\_2 \cdot \sqrt{2}\right) \cdot t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, t\_3 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, t\_3\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, t\_2, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (pow (sin y) 2.0) -0.0625))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= y -7e-5)
     (/
      (+ 2.0 (* (* t_2 (sqrt 2.0)) t_0))
      (fma 1.5 (fma (cos x) t_1 (* t_3 (cos y))) 3.0))
     (if (<= y 1.72e-6)
       (/
        (+ 2.0 (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_0))
        (fma 1.5 (fma (cos x) t_1 t_3) 3.0))
       (/
        (fma (* t_0 (sqrt 2.0)) t_2 2.0)
        (fma
         1.5
         (fma (cos x) t_1 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = pow(sin(y), 2.0) * -0.0625;
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -7e-5) {
		tmp = (2.0 + ((t_2 * sqrt(2.0)) * t_0)) / fma(1.5, fma(cos(x), t_1, (t_3 * cos(y))), 3.0);
	} else if (y <= 1.72e-6) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma(1.5, fma(cos(x), t_1, t_3), 3.0);
	} else {
		tmp = fma((t_0 * sqrt(2.0)), t_2, 2.0) / fma(1.5, fma(cos(x), t_1, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64((sin(y) ^ 2.0) * -0.0625)
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -7e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_2 * sqrt(2.0)) * t_0)) / fma(1.5, fma(cos(x), t_1, Float64(t_3 * cos(y))), 3.0));
	elseif (y <= 1.72e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(1.5, fma(cos(x), t_1, t_3), 3.0));
	else
		tmp = Float64(fma(Float64(t_0 * sqrt(2.0)), t_2, 2.0) / fma(1.5, fma(cos(x), t_1, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	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] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e-5], N[(N[(2.0 + N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.72e-6], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.9999999999999994e-5

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 61.0

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

   8. Applied rewrites 61.0%

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

   if -6.9999999999999994e-5 < y < 1.72e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

   if 1.72e-6 < y

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Applied rewrites 99.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 58.8%

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

Alternative 12: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -7 \cdot 10^{-5} \lor \neg \left(y \leq 1.72 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot t\_1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, t\_0\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (or (<= y -7e-5) (not (<= y 1.72e-6)))
     (/
      (+ 2.0 (* (* (* (pow (sin y) 2.0) -0.0625) (sqrt 2.0)) t_1))
      (fma 1.5 (fma (cos x) t_2 (* t_0 (cos y))) 3.0))
     (/
      (+ 2.0 (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_1))
      (fma 1.5 (fma (cos x) t_2 t_0) 3.0)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if ((y <= -7e-5) || !(y <= 1.72e-6)) {
		tmp = (2.0 + (((pow(sin(y), 2.0) * -0.0625) * sqrt(2.0)) * t_1)) / fma(1.5, fma(cos(x), t_2, (t_0 * cos(y))), 3.0);
	} else {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_1)) / fma(1.5, fma(cos(x), t_2, t_0), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((y <= -7e-5) || !(y <= 1.72e-6))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * sqrt(2.0)) * t_1)) / fma(1.5, fma(cos(x), t_2, Float64(t_0 * cos(y))), 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1)) / fma(1.5, fma(cos(x), t_2, t_0), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -7e-5], N[Not[LessEqual[y, 1.72e-6]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.9999999999999994e-5 or 1.72e-6 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 59.8

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

   8. Applied rewrites 59.8%

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

   if -6.9999999999999994e-5 < y < 1.72e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-5} \lor \neg \left(y \leq 1.72 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ \mathbf{if}\;y \leq -5200000 \lor \neg \left(y \leq 1.72 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\frac{\mathsf{fma}\left(t\_0, \cos x, t\_1\right)}{2} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (* (- 3.0 (sqrt 5.0)) (cos y))))
   (if (or (<= y -5200000.0) (not (<= y 1.72e-6)))
     (/
      (+
       2.0
       (* (* (* (pow (sin y) 2.0) -0.0625) (sqrt 2.0)) (- (cos x) (cos y))))
      (fma 1.5 (fma (cos x) t_0 t_1) 3.0))
     (/
      (fma
       -0.020833333333333332
       (* (* (- (cos x) 1.0) (sqrt 2.0)) (pow (sin x) 2.0))
       0.6666666666666666)
      (+ (/ (fma t_0 (cos x) t_1) 2.0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (3.0 - sqrt(5.0)) * cos(y);
	double tmp;
	if ((y <= -5200000.0) || !(y <= 1.72e-6)) {
		tmp = (2.0 + (((pow(sin(y), 2.0) * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y)))) / fma(1.5, fma(cos(x), t_0, t_1), 3.0);
	} else {
		tmp = fma(-0.020833333333333332, (((cos(x) - 1.0) * sqrt(2.0)) * pow(sin(x), 2.0)), 0.6666666666666666) / ((fma(t_0, cos(x), t_1) / 2.0) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	tmp = 0.0
	if ((y <= -5200000.0) || !(y <= 1.72e-6))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / fma(1.5, fma(cos(x), t_0, t_1), 3.0));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * (sin(x) ^ 2.0)), 0.6666666666666666) / Float64(Float64(fma(t_0, cos(x), t_1) / 2.0) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -5200000.0], N[Not[LessEqual[y, 1.72e-6]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2e6 or 1.72e-6 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 60.2

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

   8. Applied rewrites 60.2%

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

   if -5.2e6 < y < 1.72e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 98.7

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

   5. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{48}} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{48}\right), {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{48}}, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \color{blue}{{\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      16. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\color{blue}{\sin x}}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      17. metadata-eval 98.9

          \[\leadsto \frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \color{blue}{0.6666666666666666}\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

   10. Applied rewrites 98.9%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5200000 \lor \neg \left(y \leq 1.72 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_2 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1\right), 3\right)\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.00075:\\ \;\;\;\;\frac{2 + \left(\left(t\_3 \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{t\_2}\\ \mathbf{elif}\;x \leq 0.00066:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot t\_3, 0.6666666666666666\right)}{\frac{\mathsf{fma}\left(t\_0, \cos x, t\_1\right)}{2} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_2 (fma 1.5 (fma (cos x) t_0 t_1) 3.0))
        (t_3 (pow (sin x) 2.0)))
   (if (<= x -0.00075)
     (/ (+ 2.0 (* (* (* t_3 -0.0625) (sqrt 2.0)) (- (cos x) (cos y)))) t_2)
     (if (<= x 0.00066)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        t_2)
       (/
        (fma
         -0.020833333333333332
         (* (* (- (cos x) 1.0) (sqrt 2.0)) t_3)
         0.6666666666666666)
        (+ (/ (fma t_0 (cos x) t_1) 2.0) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (3.0 - sqrt(5.0)) * cos(y);
	double t_2 = fma(1.5, fma(cos(x), t_0, t_1), 3.0);
	double t_3 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -0.00075) {
		tmp = (2.0 + (((t_3 * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y)))) / t_2;
	} else if (x <= 0.00066) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_2;
	} else {
		tmp = fma(-0.020833333333333332, (((cos(x) - 1.0) * sqrt(2.0)) * t_3), 0.6666666666666666) / ((fma(t_0, cos(x), t_1) / 2.0) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_2 = fma(1.5, fma(cos(x), t_0, t_1), 3.0)
	t_3 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -0.00075)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(t_3 * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / t_2);
	elseif (x <= 0.00066)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_2);
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * t_3), 0.6666666666666666) / Float64(Float64(fma(t_0, cos(x), t_1) / 2.0) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -0.00075], N[(N[(2.0 + N[(N[(N[(t$95$3 * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 0.00066], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5000000000000002e-4

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 63.4

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

   8. Applied rewrites 63.4%

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

   if -7.5000000000000002e-4 < x < 6.6e-4

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if 6.6e-4 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 63.4

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

   5. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 63.5%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{48}} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{48}\right), {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{48}}, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \color{blue}{{\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      16. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\color{blue}{\sin x}}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      17. metadata-eval 63.7

          \[\leadsto \frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \color{blue}{0.6666666666666666}\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

   10. Applied rewrites 63.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_1 := \mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -160000000:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, t\_0\right), 3\right)}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\frac{\mathsf{fma}\left(t\_2, \cos x, t\_0\right)}{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_1
         (fma
          (* (pow (sin y) 2.0) -0.0625)
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= y -160000000.0)
     (/ t_1 (fma 1.5 (fma (cos x) t_2 t_0) 3.0))
     (if (<= y 1.72e-6)
       (/
        (fma
         -0.020833333333333332
         (* (* (- (cos x) 1.0) (sqrt 2.0)) (pow (sin x) 2.0))
         0.6666666666666666)
        (+ (/ (fma t_2 (cos x) t_0) 2.0) 1.0))
       (/
        t_1
        (fma
         1.5
         (fma (cos x) t_2 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) * cos(y);
	double t_1 = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -160000000.0) {
		tmp = t_1 / fma(1.5, fma(cos(x), t_2, t_0), 3.0);
	} else if (y <= 1.72e-6) {
		tmp = fma(-0.020833333333333332, (((cos(x) - 1.0) * sqrt(2.0)) * pow(sin(x), 2.0)), 0.6666666666666666) / ((fma(t_2, cos(x), t_0) / 2.0) + 1.0);
	} else {
		tmp = t_1 / fma(1.5, fma(cos(x), t_2, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_1 = fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -160000000.0)
		tmp = Float64(t_1 / fma(1.5, fma(cos(x), t_2, t_0), 3.0));
	elseif (y <= 1.72e-6)
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * (sin(x) ^ 2.0)), 0.6666666666666666) / Float64(Float64(fma(t_2, cos(x), t_0) / 2.0) + 1.0));
	else
		tmp = Float64(t_1 / fma(1.5, fma(cos(x), t_2, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -160000000.0], N[(t$95$1 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.72e-6], N[(N[(-0.020833333333333332 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6e8

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -1.6e8 < y < 1.72e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 98.2

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

   5. Applied rewrites 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{48}} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{48}\right), {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{48}}, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \color{blue}{{\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      16. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\color{blue}{\sin x}}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      17. metadata-eval 98.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \color{blue}{0.6666666666666666}\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

   10. Applied rewrites 98.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

   if 1.72e-6 < y

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

      12. lower-sqrt.f64 58.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Applied rewrites 58.6%

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

Alternative 16: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ \mathbf{if}\;y \leq -160000000 \lor \neg \left(y \leq 1.72 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\frac{\mathsf{fma}\left(t\_0, \cos x, t\_1\right)}{2} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (* (- 3.0 (sqrt 5.0)) (cos y))))
   (if (or (<= y -160000000.0) (not (<= y 1.72e-6)))
     (/
      (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma 1.5 (fma (cos x) t_0 t_1) 3.0))
     (/
      (fma
       -0.020833333333333332
       (* (* (- (cos x) 1.0) (sqrt 2.0)) (pow (sin x) 2.0))
       0.6666666666666666)
      (+ (/ (fma t_0 (cos x) t_1) 2.0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (3.0 - sqrt(5.0)) * cos(y);
	double tmp;
	if ((y <= -160000000.0) || !(y <= 1.72e-6)) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, t_1), 3.0);
	} else {
		tmp = fma(-0.020833333333333332, (((cos(x) - 1.0) * sqrt(2.0)) * pow(sin(x), 2.0)), 0.6666666666666666) / ((fma(t_0, cos(x), t_1) / 2.0) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	tmp = 0.0
	if ((y <= -160000000.0) || !(y <= 1.72e-6))
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, t_1), 3.0));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * (sin(x) ^ 2.0)), 0.6666666666666666) / Float64(Float64(fma(t_0, cos(x), t_1) / 2.0) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -160000000.0], N[Not[LessEqual[y, 1.72e-6]], $MachinePrecision]], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e8 or 1.72e-6 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 60.3

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

   8. Applied rewrites 60.3%

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

   if -1.6e8 < y < 1.72e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 98.2

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

   5. Applied rewrites 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{48}} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \frac{1}{3} \cdot 2}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{48}\right), {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{48}}, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot {\sin x}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \color{blue}{{\sin x}^{2}}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      16. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{48}, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\color{blue}{\sin x}}^{2}, \frac{1}{3} \cdot 2\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

      17. metadata-eval 98.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, \color{blue}{0.6666666666666666}\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]

   10. Applied rewrites 98.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -160000000 \lor \neg \left(y \leq 1.72 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-6} \lor \neg \left(y \leq 1.72 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= y -4.4e-6) (not (<= y 1.72e-6)))
     (/
      (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma 1.5 (fma (cos x) t_0 (* t_1 (cos y))) 3.0))
     (/
      (fma
       -0.020833333333333332
       (* (* (- (cos x) 1.0) (sqrt 2.0)) (pow (sin x) 2.0))
       0.6666666666666666)
      (fma (fma t_0 (cos x) t_1) 0.5 1.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((y <= -4.4e-6) || !(y <= 1.72e-6)) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, (t_1 * cos(y))), 3.0);
	} else {
		tmp = fma(-0.020833333333333332, (((cos(x) - 1.0) * sqrt(2.0)) * pow(sin(x), 2.0)), 0.6666666666666666) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((y <= -4.4e-6) || !(y <= 1.72e-6))
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(t_1 * cos(y))), 3.0));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * (sin(x) ^ 2.0)), 0.6666666666666666) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -4.4e-6], N[Not[LessEqual[y, 1.72e-6]], $MachinePrecision]], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4000000000000002e-6 or 1.72e-6 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 59.7

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

   8. Applied rewrites 59.7%

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

   if -4.4000000000000002e-6 < y < 1.72e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-6} \lor \neg \left(y \leq 1.72 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00075 \lor \neg \left(x \leq 0.0007\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.00075) (not (<= x 0.0007)))
     (/
      (fma (* (pow (sin x) 2.0) -0.0625) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
      (fma 1.5 (fma (cos x) t_0 (* t_1 (cos y))) 3.0))
     (/
      (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma
       (* t_0 (fma 0.0625 (* x x) -0.75))
       (* x x)
       (fma 1.5 (fma t_1 (cos y) t_0) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.00075) || !(x <= 0.0007)) {
		tmp = fma((pow(sin(x), 2.0) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, (t_1 * cos(y))), 3.0);
	} else {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((t_0 * fma(0.0625, (x * x), -0.75)), (x * x), fma(1.5, fma(t_1, cos(y), t_0), 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.00075) || !(x <= 0.0007))
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(t_1 * cos(y))), 3.0));
	else
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(t_0 * fma(0.0625, Float64(x * x), -0.75)), Float64(x * x), fma(1.5, fma(t_1, cos(y), t_0), 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00075], N[Not[LessEqual[x, 0.0007]], $MachinePrecision]], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[(0.0625 * N[(x * x), $MachinePrecision] + -0.75), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000002e-4 or 6.99999999999999993e-4 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 63.4

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

   8. Applied rewrites 63.4%

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

   if -7.5000000000000002e-4 < x < 6.99999999999999993e-4

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right), {x}^{2}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      12. lower-sqrt.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   8. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00075 \lor \neg \left(x \leq 0.0007\right):\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 78.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := {\sin x}^{2}\\ t_3 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot -0.0625, t\_3, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, \frac{4}{\sqrt{5} + 3}\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.00105:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(t\_1 \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos y, t\_1\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, t\_3 \cdot t\_2, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_0\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

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))
        (t_3 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -0.00125)
     (/
      (fma (* t_2 -0.0625) t_3 2.0)
      (fma 1.5 (fma (cos x) t_1 (/ 4.0 (+ (sqrt 5.0) 3.0))) 3.0))
     (if (<= x 0.00105)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma
         (* t_1 (fma 0.0625 (* x x) -0.75))
         (* x x)
         (fma 1.5 (fma t_0 (cos y) t_1) 3.0)))
       (/
        (fma -0.020833333333333332 (* t_3 t_2) 0.6666666666666666)
        (fma (fma t_1 (cos x) t_0) 0.5 1.0))))))

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 t_3 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -0.00125) {
		tmp = fma((t_2 * -0.0625), t_3, 2.0) / fma(1.5, fma(cos(x), t_1, (4.0 / (sqrt(5.0) + 3.0))), 3.0);
	} else if (x <= 0.00105) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((t_1 * fma(0.0625, (x * x), -0.75)), (x * x), fma(1.5, fma(t_0, cos(y), t_1), 3.0));
	} else {
		tmp = fma(-0.020833333333333332, (t_3 * t_2), 0.6666666666666666) / fma(fma(t_1, cos(x), t_0), 0.5, 1.0);
	}
	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
	t_3 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -0.00125)
		tmp = Float64(fma(Float64(t_2 * -0.0625), t_3, 2.0) / fma(1.5, fma(cos(x), t_1, Float64(4.0 / Float64(sqrt(5.0) + 3.0))), 3.0));
	elseif (x <= 0.00105)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(t_1 * fma(0.0625, Float64(x * x), -0.75)), Float64(x * x), fma(1.5, fma(t_0, cos(y), t_1), 3.0)));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(t_3 * t_2), 0.6666666666666666) / fma(fma(t_1, cos(x), t_0), 0.5, 1.0));
	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]}, Block[{t$95$3 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00125], N[(N[(N[(t$95$2 * -0.0625), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00105], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[(0.0625 * N[(x * x), $MachinePrecision] + -0.75), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(1.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(t$95$3 * t$95$2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00125000000000000003

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 61.8

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

   8. Applied rewrites 61.8%

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

   10. Applied rewrites 61.9%

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3}\right), 3\right)} \]

   if -0.00125000000000000003 < x < 0.00104999999999999994

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right), {x}^{2}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{16}, x \cdot x, \frac{-3}{4}\right), x \cdot x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      12. lower-sqrt.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   8. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \mathsf{fma}\left(0.0625, x \cdot x, -0.75\right), x \cdot x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   if 0.00104999999999999994 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 63.4

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

   5. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 63.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 62.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 77.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-6} \lor \neg \left(x \leq 2 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -3.6e-6) (not (<= x 2e-39)))
     (/
      (fma
       -0.020833333333333332
       (* (* (- (cos x) 1.0) (sqrt 2.0)) (pow (sin x) 2.0))
       0.6666666666666666)
      (fma (fma t_0 (cos x) t_1) 0.5 1.0))
     (*
      (/
       (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
       (fma 0.5 (fma (cos y) t_1 t_0) 1.0))
      0.3333333333333333))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -3.6e-6) || !(x <= 2e-39)) {
		tmp = fma(-0.020833333333333332, (((cos(x) - 1.0) * sqrt(2.0)) * pow(sin(x), 2.0)), 0.6666666666666666) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0);
	} else {
		tmp = (fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_1, t_0), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -3.6e-6) || !(x <= 2e-39))
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * (sin(x) ^ 2.0)), 0.6666666666666666) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0));
	else
		tmp = Float64(Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_1, t_0), 1.0)) * 0.3333333333333333);
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -3.6e-6], N[Not[LessEqual[x, 2e-39]], $MachinePrecision]], N[(N[(-0.020833333333333332 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.59999999999999984e-6 or 1.99999999999999986e-39 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 63.9

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

   5. Applied rewrites 63.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 62.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]

   if -3.59999999999999984e-6 < x < 1.99999999999999986e-39

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-6} \lor \neg \left(x \leq 2 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 77.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin x}^{2}\\ t_2 := 3 - \sqrt{5}\\ t_3 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot -0.0625, t\_3, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \frac{4}{\sqrt{5} + 3}\right), 3\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_2, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, t\_3 \cdot t\_1, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_2\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -8e-7)
     (/
      (fma (* t_1 -0.0625) t_3 2.0)
      (fma 1.5 (fma (cos x) t_0 (/ 4.0 (+ (sqrt 5.0) 3.0))) 3.0))
     (if (<= x 2e-39)
       (*
        (/
         (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
         (fma 0.5 (fma (cos y) t_2 t_0) 1.0))
        0.3333333333333333)
       (/
        (fma -0.020833333333333332 (* t_3 t_1) 0.6666666666666666)
        (fma (fma t_0 (cos x) t_2) 0.5 1.0))))))

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 = 3.0 - sqrt(5.0);
	double t_3 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -8e-7) {
		tmp = fma((t_1 * -0.0625), t_3, 2.0) / fma(1.5, fma(cos(x), t_0, (4.0 / (sqrt(5.0) + 3.0))), 3.0);
	} else if (x <= 2e-39) {
		tmp = (fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_2, t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma(-0.020833333333333332, (t_3 * t_1), 0.6666666666666666) / fma(fma(t_0, cos(x), t_2), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -8e-7)
		tmp = Float64(fma(Float64(t_1 * -0.0625), t_3, 2.0) / fma(1.5, fma(cos(x), t_0, Float64(4.0 / Float64(sqrt(5.0) + 3.0))), 3.0));
	elseif (x <= 2e-39)
		tmp = Float64(Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_2, t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(t_3 * t_1), 0.6666666666666666) / fma(fma(t_0, cos(x), t_2), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-7], N[(N[(N[(t$95$1 * -0.0625), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-39], N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(t$95$3 * t$95$1), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.9999999999999996e-7

   1. Initial program 98.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 63.1

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 63.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

      13. lower-sqrt.f64 61.8

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 61.8%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.9%

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3}\right), 3\right)} \]

   if -7.9999999999999996e-7 < x < 1.99999999999999986e-39

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333} \]

   if 1.99999999999999986e-39 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 64.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 64.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3}}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y}} \]

   7. Applied rewrites 64.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}} \]

   10. Applied rewrites 63.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3}\right), 3\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 77.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-6} \lor \neg \left(x \leq 2 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos y, t\_0\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -3.6e-6) (not (<= x 2e-39)))
     (/
      (fma
       -0.020833333333333332
       (* (* (- (cos x) 1.0) (sqrt 2.0)) (pow (sin x) 2.0))
       0.6666666666666666)
      (fma (fma t_0 (cos x) t_1) 0.5 1.0))
     (/
      (fma
       -0.020833333333333332
       (* (* (- 1.0 (cos y)) (sqrt 2.0)) (pow (sin y) 2.0))
       0.6666666666666666)
      (fma (fma t_1 (cos y) t_0) 0.5 1.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -3.6e-6) || !(x <= 2e-39)) {
		tmp = fma(-0.020833333333333332, (((cos(x) - 1.0) * sqrt(2.0)) * pow(sin(x), 2.0)), 0.6666666666666666) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0);
	} else {
		tmp = fma(-0.020833333333333332, (((1.0 - cos(y)) * sqrt(2.0)) * pow(sin(y), 2.0)), 0.6666666666666666) / fma(fma(t_1, cos(y), t_0), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -3.6e-6) || !(x <= 2e-39))
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * (sin(x) ^ 2.0)), 0.6666666666666666) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0));
	else
		tmp = Float64(fma(-0.020833333333333332, Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * (sin(y) ^ 2.0)), 0.6666666666666666) / fma(fma(t_1, cos(y), t_0), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -3.6e-6], N[Not[LessEqual[x, 2e-39]], $MachinePrecision]], N[(N[(-0.020833333333333332 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.59999999999999984e-6 or 1.99999999999999986e-39 < x

   1. Initial program 98.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 63.9

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 63.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3}}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y}} \]

   7. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}} \]

   10. Applied rewrites 62.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]

   if -3.59999999999999984e-6 < x < 1.99999999999999986e-39

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower-sqrt.f64 67.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 67.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3}}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y}} \]

   7. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{3}}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}} \]

   10. Applied rewrites 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 0.5, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-6} \lor \neg \left(x \leq 2 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot {\sin x}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.020833333333333332, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot {\sin y}^{2}, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 45.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{3 \cdot \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 - \frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   3.0
   (fma
    (/ (- (sqrt 5.0) 1.0) 2.0)
    (cos x)
    (- 1.0 (* (/ (- 3.0 (sqrt 5.0)) -2.0) (cos y)))))))

C

double code(double x, double y) {
	return 2.0 / (3.0 * fma(((sqrt(5.0) - 1.0) / 2.0), cos(x), (1.0 - (((3.0 - sqrt(5.0)) / -2.0) * cos(y)))));
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(3.0 * fma(Float64(Float64(sqrt(5.0) - 1.0) / 2.0), cos(x), Float64(1.0 - Float64(Float64(Float64(3.0 - sqrt(5.0)) / -2.0) * cos(y))))))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(3.0 * N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(1.0 - N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. lower-sqrt.f64 65.5

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 65.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)}} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

   6. associate--l+ N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \left(1 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)}} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \color{blue}{1 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y}\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 - \color{blue}{\left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y}\right)} \]

7. Applied rewrites 65.5%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 - \frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 - \frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 46.9%

    \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1 - \frac{3 - \sqrt{5}}{-2} \cdot \cos y\right)} \]
11. Add Preprocessing

Alternative 24: 45.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   1.5
   (fma (- 3.0 (sqrt 5.0)) (cos y) (* (- (sqrt 5.0) 1.0) (cos x)))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma((3.0 - sqrt(5.0)), cos(y), ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. lower-sqrt.f64 65.5

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 65.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   13. lower-sqrt.f64 63.4

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 63.4%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 44.7%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

12. Taylor expanded in x around inf

    \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
13. Step-by-step derivation

    1. distribute-lft-in N/A

       \[\leadsto \frac{2}{3 \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

    2. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

    3. distribute-lft-in N/A

       \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

    4. associate-*r* N/A

       \[\leadsto \frac{2}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

    5. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

    6. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

    7. metadata-eval N/A

       \[\leadsto \frac{2}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

    8. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

14. Applied rewrites 46.9%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}} \]
15. Add Preprocessing

Alternative 25: 43.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. lower-sqrt.f64 65.5

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 65.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   13. lower-sqrt.f64 63.4

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 63.4%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 44.7%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025008 
(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))))))