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

Percentage Accurate: 99.3% → 99.3%

Time: 13.2s

Alternatives: 33

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift--.f64 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in y around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 99.3

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

10. Applied rewrites 99.3%

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

12. Applied rewrites 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift--.f64 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in y around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 99.3

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

10. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift--.f64 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-fma.f64 N/A

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

10. Applied rewrites 99.3%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (* (sqrt 2.0) (- (cos x) (cos y)))
    (* (- (sin y) (* 0.0625 (sin x))) (- (sin x) (* 0.0625 (sin y))))
    2.0)
   (fma
    0.5
    (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
    1.0))
  0.3333333333333333))

C

double code(double x, double y) {
	return (fma((sqrt(2.0) * (cos(x) - cos(y))), ((sin(y) - (0.0625 * sin(x))) * (sin(x) - (0.0625 * sin(y)))), 2.0) / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)) * 0.3333333333333333)
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[(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[(0.5 * 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] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 99.2%

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

Alternative 5: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \cos x - \cos y\\ t_3 := 3 - \sqrt{5}\\ t_4 := \frac{t\_3}{2}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_1\right) \cdot t\_2}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \left(0.5 \cdot \cos y\right) \cdot t\_3\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right) + t\_4 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\sin x \cdot \sqrt{2}\right) \cdot t\_1, t\_2, 2\right)}{3}}{\mathsf{fma}\left(\cos y, t\_4, \mathsf{fma}\left(\frac{\sqrt{5}}{2} - 0.5, \cos x, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (sin y) (/ (sin x) 16.0)))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (/ t_3 2.0)))
   (if (<= x -0.31)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) t_1) t_2))
      (* 3.0 (+ (+ 1.0 (* (/ t_0 2.0) (cos x))) (* (* 0.5 (cos y)) t_3))))
     (if (<= x 5.8e-7)
       (/
        (fma
         (*
          (sqrt 2.0)
          (-
           (fma
            (-
             (*
              (fma -0.001388888888888889 (* x x) 0.041666666666666664)
              (* x x))
             0.5)
            (* x x)
            1.0)
           (cos y)))
         (*
          (fma
           (-
            (*
             (fma -0.0005208333333333333 (* x x) 0.010416666666666666)
             (* x x))
            0.0625)
           x
           (sin y))
          (- (sin x) (* 0.0625 (sin y))))
         2.0)
        (* 3.0 (+ (fma (* 0.5 (cos x)) t_0 1.0) (* t_4 (cos y)))))
       (/
        (/ (fma (* (* (sin x) (sqrt 2.0)) t_1) t_2 2.0) 3.0)
        (fma (cos y) t_4 (fma (- (/ (sqrt 5.0) 2.0) 0.5) (cos x) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = sin(y) - (sin(x) / 16.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 / 2.0;
	double tmp;
	if (x <= -0.31) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * t_1) * t_2)) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + ((0.5 * cos(y)) * t_3)));
	} else if (x <= 5.8e-7) {
		tmp = fma((sqrt(2.0) * (fma(((fma(-0.001388888888888889, (x * x), 0.041666666666666664) * (x * x)) - 0.5), (x * x), 1.0) - cos(y))), (fma(((fma(-0.0005208333333333333, (x * x), 0.010416666666666666) * (x * x)) - 0.0625), x, sin(y)) * (sin(x) - (0.0625 * sin(y)))), 2.0) / (3.0 * (fma((0.5 * cos(x)), t_0, 1.0) + (t_4 * cos(y))));
	} else {
		tmp = (fma(((sin(x) * sqrt(2.0)) * t_1), t_2, 2.0) / 3.0) / fma(cos(y), t_4, fma(((sqrt(5.0) / 2.0) - 0.5), cos(x), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 / 2.0)
	tmp = 0.0
	if (x <= -0.31)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * t_1) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(0.5 * cos(y)) * t_3))));
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(fma(Float64(Float64(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664) * Float64(x * x)) - 0.5), Float64(x * x), 1.0) - cos(y))), Float64(fma(Float64(Float64(fma(-0.0005208333333333333, Float64(x * x), 0.010416666666666666) * Float64(x * x)) - 0.0625), x, sin(y)) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / Float64(3.0 * Float64(fma(Float64(0.5 * cos(x)), t_0, 1.0) + Float64(t_4 * cos(y)))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(sin(x) * sqrt(2.0)) * t_1), t_2, 2.0) / 3.0) / fma(cos(y), t_4, fma(Float64(Float64(sqrt(5.0) / 2.0) - 0.5), cos(x), 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[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.31], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0005208333333333333 * N[(x * x), $MachinePrecision] + 0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] + N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * t$95$4 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.309999999999999998

   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. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.9

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 62.9

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

   7. Applied rewrites 62.9%

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

   if -0.309999999999999998 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y + x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right)\right) \cdot \left(\color{blue}{\sin x} - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) \cdot x + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {x}^{2} + \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, {x}^{2}, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-sin.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\color{blue}{\sin x} - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       11. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       13. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       14. lift-*.f64 99.6

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in y around 0

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

      1. lift-sin.f64 63.2

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

   4. Applied rewrites 63.2%

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

   6. Applied rewrites 63.2%

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

Alternative 6: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right)\\ t_2 := \cos x - \cos y\\ t_3 := 3 - \sqrt{5}\\ t_4 := \left(0.5 \cdot \cos y\right) \cdot t\_3\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_2}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + t\_4\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(t\_1 + \frac{t\_3}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_2, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \sin x, 2\right)}{3 \cdot \left(t\_1 + t\_4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (fma (* 0.5 (cos x)) t_0 1.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (* (* 0.5 (cos y)) t_3)))
   (if (<= x -0.31)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0))) t_2))
      (* 3.0 (+ (+ 1.0 (* (/ t_0 2.0) (cos x))) t_4)))
     (if (<= x 5.8e-7)
       (/
        (fma
         (*
          (sqrt 2.0)
          (-
           (fma
            (-
             (*
              (fma -0.001388888888888889 (* x x) 0.041666666666666664)
              (* x x))
             0.5)
            (* x x)
            1.0)
           (cos y)))
         (*
          (fma
           (-
            (*
             (fma -0.0005208333333333333 (* x x) 0.010416666666666666)
             (* x x))
            0.0625)
           x
           (sin y))
          (- (sin x) (* 0.0625 (sin y))))
         2.0)
        (* 3.0 (+ t_1 (* (/ t_3 2.0) (cos y)))))
       (/
        (fma (* (sqrt 2.0) t_2) (* (- (sin y) (* 0.0625 (sin x))) (sin x)) 2.0)
        (* 3.0 (+ t_1 t_4)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma((0.5 * cos(x)), t_0, 1.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = (0.5 * cos(y)) * t_3;
	double tmp;
	if (x <= -0.31) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * t_2)) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + t_4));
	} else if (x <= 5.8e-7) {
		tmp = fma((sqrt(2.0) * (fma(((fma(-0.001388888888888889, (x * x), 0.041666666666666664) * (x * x)) - 0.5), (x * x), 1.0) - cos(y))), (fma(((fma(-0.0005208333333333333, (x * x), 0.010416666666666666) * (x * x)) - 0.0625), x, sin(y)) * (sin(x) - (0.0625 * sin(y)))), 2.0) / (3.0 * (t_1 + ((t_3 / 2.0) * cos(y))));
	} else {
		tmp = fma((sqrt(2.0) * t_2), ((sin(y) - (0.0625 * sin(x))) * sin(x)), 2.0) / (3.0 * (t_1 + t_4));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(Float64(0.5 * cos(x)), t_0, 1.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(Float64(0.5 * cos(y)) * t_3)
	tmp = 0.0
	if (x <= -0.31)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + t_4)));
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(fma(Float64(Float64(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664) * Float64(x * x)) - 0.5), Float64(x * x), 1.0) - cos(y))), Float64(fma(Float64(Float64(fma(-0.0005208333333333333, Float64(x * x), 0.010416666666666666) * Float64(x * x)) - 0.0625), x, sin(y)) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / Float64(3.0 * Float64(t_1 + Float64(Float64(t_3 / 2.0) * cos(y)))));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * t_2), Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * sin(x)), 2.0) / Float64(3.0 * Float64(t_1 + t_4)));
	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[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[x, -0.31], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0005208333333333333 * N[(x * x), $MachinePrecision] + 0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$1 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.309999999999999998

   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. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.9

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 62.9

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

   7. Applied rewrites 62.9%

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

   if -0.309999999999999998 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y + x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right)\right) \cdot \left(\color{blue}{\sin x} - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) \cdot x + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {x}^{2} + \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, {x}^{2}, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-sin.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\color{blue}{\sin x} - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       11. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       13. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       14. lift-*.f64 99.6

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in y around 0

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

       1. lift-sin.f64 63.2

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

   13. Applied rewrites 63.2%

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

Alternative 7: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right)\\ t_3 := 3 \cdot \left(t\_2 + \frac{t\_0}{2} \cdot \cos y\right)\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{t\_3}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_1, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \sin x, 2\right)}{3 \cdot \left(t\_2 + \left(0.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (fma (* 0.5 (cos x)) (- (sqrt 5.0) 1.0) 1.0))
        (t_3 (* 3.0 (+ t_2 (* (/ t_0 2.0) (cos y))))))
   (if (<= x -0.31)
     (/
      (+ 2.0 (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_1))
      t_3)
     (if (<= x 5.8e-7)
       (/
        (fma
         (*
          (sqrt 2.0)
          (-
           (fma
            (-
             (*
              (fma -0.001388888888888889 (* x x) 0.041666666666666664)
              (* x x))
             0.5)
            (* x x)
            1.0)
           (cos y)))
         (*
          (fma
           (-
            (*
             (fma -0.0005208333333333333 (* x x) 0.010416666666666666)
             (* x x))
            0.0625)
           x
           (sin y))
          (- (sin x) (* 0.0625 (sin y))))
         2.0)
        t_3)
       (/
        (fma (* (sqrt 2.0) t_1) (* (- (sin y) (* 0.0625 (sin x))) (sin x)) 2.0)
        (* 3.0 (+ t_2 (* (* 0.5 (cos y)) t_0))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = fma((0.5 * cos(x)), (sqrt(5.0) - 1.0), 1.0);
	double t_3 = 3.0 * (t_2 + ((t_0 / 2.0) * cos(y)));
	double tmp;
	if (x <= -0.31) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_1)) / t_3;
	} else if (x <= 5.8e-7) {
		tmp = fma((sqrt(2.0) * (fma(((fma(-0.001388888888888889, (x * x), 0.041666666666666664) * (x * x)) - 0.5), (x * x), 1.0) - cos(y))), (fma(((fma(-0.0005208333333333333, (x * x), 0.010416666666666666) * (x * x)) - 0.0625), x, sin(y)) * (sin(x) - (0.0625 * sin(y)))), 2.0) / t_3;
	} else {
		tmp = fma((sqrt(2.0) * t_1), ((sin(y) - (0.0625 * sin(x))) * sin(x)), 2.0) / (3.0 * (t_2 + ((0.5 * cos(y)) * t_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 = fma(Float64(0.5 * cos(x)), Float64(sqrt(5.0) - 1.0), 1.0)
	t_3 = Float64(3.0 * Float64(t_2 + Float64(Float64(t_0 / 2.0) * cos(y))))
	tmp = 0.0
	if (x <= -0.31)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1)) / t_3);
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(fma(Float64(Float64(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664) * Float64(x * x)) - 0.5), Float64(x * x), 1.0) - cos(y))), Float64(fma(Float64(Float64(fma(-0.0005208333333333333, Float64(x * x), 0.010416666666666666) * Float64(x * x)) - 0.0625), x, sin(y)) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / t_3);
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * t_1), Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * sin(x)), 2.0) / Float64(3.0 * Float64(t_2 + Float64(Float64(0.5 * cos(y)) * t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 * N[(t$95$2 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], 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] / t$95$3), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0005208333333333333 * N[(x * x), $MachinePrecision] + 0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.309999999999999998

   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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 62.9

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

   7. Applied rewrites 62.9%

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

   if -0.309999999999999998 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y + x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right)\right) \cdot \left(\color{blue}{\sin x} - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) \cdot x + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {x}^{2} + \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, {x}^{2}, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-sin.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\color{blue}{\sin x} - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       11. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       13. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       14. lift-*.f64 99.6

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in y around 0

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

       1. lift-sin.f64 63.2

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

   13. Applied rewrites 63.2%

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

Alternative 8: 81.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right)\\ t_2 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \sin x, 2\right)}{3 \cdot \left(t\_1 + \left(0.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(t\_1 + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (* 0.5 (cos x)) (- (sqrt 5.0) 1.0) 1.0))
        (t_2
         (/
          (fma
           (* (sqrt 2.0) (- (cos x) (cos y)))
           (* (- (sin y) (* 0.0625 (sin x))) (sin x))
           2.0)
          (* 3.0 (+ t_1 (* (* 0.5 (cos y)) t_0))))))
   (if (<= x -0.31)
     t_2
     (if (<= x 5.8e-7)
       (/
        (fma
         (*
          (sqrt 2.0)
          (-
           (fma
            (-
             (*
              (fma -0.001388888888888889 (* x x) 0.041666666666666664)
              (* x x))
             0.5)
            (* x x)
            1.0)
           (cos y)))
         (*
          (fma
           (-
            (*
             (fma -0.0005208333333333333 (* x x) 0.010416666666666666)
             (* x x))
            0.0625)
           x
           (sin y))
          (- (sin x) (* 0.0625 (sin y))))
         2.0)
        (* 3.0 (+ t_1 (* (/ t_0 2.0) (cos y)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((0.5 * cos(x)), (sqrt(5.0) - 1.0), 1.0);
	double t_2 = fma((sqrt(2.0) * (cos(x) - cos(y))), ((sin(y) - (0.0625 * sin(x))) * sin(x)), 2.0) / (3.0 * (t_1 + ((0.5 * cos(y)) * t_0)));
	double tmp;
	if (x <= -0.31) {
		tmp = t_2;
	} else if (x <= 5.8e-7) {
		tmp = fma((sqrt(2.0) * (fma(((fma(-0.001388888888888889, (x * x), 0.041666666666666664) * (x * x)) - 0.5), (x * x), 1.0) - cos(y))), (fma(((fma(-0.0005208333333333333, (x * x), 0.010416666666666666) * (x * x)) - 0.0625), x, sin(y)) * (sin(x) - (0.0625 * sin(y)))), 2.0) / (3.0 * (t_1 + ((t_0 / 2.0) * cos(y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(Float64(0.5 * cos(x)), Float64(sqrt(5.0) - 1.0), 1.0)
	t_2 = Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * sin(x)), 2.0) / Float64(3.0 * Float64(t_1 + Float64(Float64(0.5 * cos(y)) * t_0))))
	tmp = 0.0
	if (x <= -0.31)
		tmp = t_2;
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(fma(Float64(Float64(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664) * Float64(x * x)) - 0.5), Float64(x * x), 1.0) - cos(y))), Float64(fma(Float64(Float64(fma(-0.0005208333333333333, Float64(x * x), 0.010416666666666666) * Float64(x * x)) - 0.0625), x, sin(y)) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / Float64(3.0 * Float64(t_1 + Float64(Float64(t_0 / 2.0) * cos(y)))));
	else
		tmp = t_2;
	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[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], t$95$2, If[LessEqual[x, 5.8e-7], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0005208333333333333 * N[(x * x), $MachinePrecision] + 0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.309999999999999998 or 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in y around 0

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

       1. lift-sin.f64 63.1

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

   13. Applied rewrites 63.1%

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

   if -0.309999999999999998 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y + x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right)\right) \cdot \left(\color{blue}{\sin x} - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) \cdot x + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {x}^{2} + \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, {x}^{2}, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-sin.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\color{blue}{\sin x} - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       11. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       13. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       14. lift-*.f64 99.6

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 81.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right) + \left(0.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{if}\;y \leq -0.0115:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.0039:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(\mathsf{fma}\left(y \cdot \sqrt{2}, -0.0625, \left(1.00390625 \cdot \sin x\right) \cdot \sqrt{2}\right), y, \left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (fma
           (* (sqrt 2.0) (- (cos x) (cos y)))
           (* (sin y) (- (sin x) (* 0.0625 (sin y))))
           2.0)
          (* 3.0 (+ (fma (* 0.5 (cos x)) t_1 1.0) (* (* 0.5 (cos y)) t_0))))))
   (if (<= y -0.0115)
     t_2
     (if (<= y 0.0039)
       (/
        (+
         2.0
         (*
          (fma
           (fma (* y (sqrt 2.0)) -0.0625 (* (* 1.00390625 (sin x)) (sqrt 2.0)))
           y
           (* (* (pow (sin x) 2.0) -0.0625) (sqrt 2.0)))
          (- (cos x) 1.0)))
        (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))
       t_2))))

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 = fma((sqrt(2.0) * (cos(x) - cos(y))), (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / (3.0 * (fma((0.5 * cos(x)), t_1, 1.0) + ((0.5 * cos(y)) * t_0)));
	double tmp;
	if (y <= -0.0115) {
		tmp = t_2;
	} else if (y <= 0.0039) {
		tmp = (2.0 + (fma(fma((y * sqrt(2.0)), -0.0625, ((1.00390625 * sin(x)) * sqrt(2.0))), y, ((pow(sin(x), 2.0) * -0.0625) * sqrt(2.0))) * (cos(x) - 1.0))) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	} else {
		tmp = t_2;
	}
	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(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / Float64(3.0 * Float64(fma(Float64(0.5 * cos(x)), t_1, 1.0) + Float64(Float64(0.5 * cos(y)) * t_0))))
	tmp = 0.0
	if (y <= -0.0115)
		tmp = t_2;
	elseif (y <= 0.0039)
		tmp = Float64(Float64(2.0 + Float64(fma(fma(Float64(y * sqrt(2.0)), -0.0625, Float64(Float64(1.00390625 * sin(x)) * sqrt(2.0))), y, Float64(Float64((sin(x) ^ 2.0) * -0.0625) * sqrt(2.0))) * Float64(cos(x) - 1.0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	else
		tmp = t_2;
	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[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] + N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0115], t$95$2, If[LessEqual[y, 0.0039], N[(N[(2.0 + N[(N[(N[(N[(y * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625 + N[(N[(1.00390625 * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $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], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0115 or 0.0038999999999999998 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 99.1

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

   10. Applied rewrites 99.1%

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

   11. Taylor expanded in x around 0

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

       1. lift-sin.f64 63.1

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

   13. Applied rewrites 63.1%

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

   if -0.0115 < y < 0.0038999999999999998

   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. Taylor expanded in y around 0

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \sqrt{2}, -0.0625, \left(1.00390625 \cdot \sin x\right) \cdot \sqrt{2}\right), y, \left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - 1\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := \frac{t\_0}{2} \cdot \cos y\\ \mathbf{if}\;x \leq -0.43:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + t\_4\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_2, 1\right) + t\_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, t\_0\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) 1.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (* (/ t_0 2.0) (cos y))))
   (if (<= x -0.43)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) t_3) t_1))
      (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) t_4)))
     (if (<= x 5.8e-7)
       (/
        (fma
         (*
          (sqrt 2.0)
          (-
           (fma
            (-
             (*
              (fma -0.001388888888888889 (* x x) 0.041666666666666664)
              (* x x))
             0.5)
            (* x x)
            1.0)
           (cos y)))
         (*
          (fma
           (-
            (*
             (fma -0.0005208333333333333 (* x x) 0.010416666666666666)
             (* x x))
            0.0625)
           x
           (sin y))
          (- (sin x) (* 0.0625 (sin y))))
         2.0)
        (* 3.0 (+ (fma (* 0.5 (cos x)) t_2 1.0) t_4)))
       (/
        (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_3) t_1))
        (* 3.0 (fma (fma (cos x) t_2 t_0) 0.5 1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = (t_0 / 2.0) * cos(y);
	double tmp;
	if (x <= -0.43) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * t_3) * t_1)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + t_4));
	} else if (x <= 5.8e-7) {
		tmp = fma((sqrt(2.0) * (fma(((fma(-0.001388888888888889, (x * x), 0.041666666666666664) * (x * x)) - 0.5), (x * x), 1.0) - cos(y))), (fma(((fma(-0.0005208333333333333, (x * x), 0.010416666666666666) * (x * x)) - 0.0625), x, sin(y)) * (sin(x) - (0.0625 * sin(y)))), 2.0) / (3.0 * (fma((0.5 * cos(x)), t_2, 1.0) + t_4));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_3) * t_1)) / (3.0 * fma(fma(cos(x), t_2, t_0), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(Float64(t_0 / 2.0) * cos(y))
	tmp = 0.0
	if (x <= -0.43)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * t_3) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + t_4)));
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(fma(Float64(Float64(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664) * Float64(x * x)) - 0.5), Float64(x * x), 1.0) - cos(y))), Float64(fma(Float64(Float64(fma(-0.0005208333333333333, Float64(x * x), 0.010416666666666666) * Float64(x * x)) - 0.0625), x, sin(y)) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / Float64(3.0 * Float64(fma(Float64(0.5 * cos(x)), t_2, 1.0) + t_4)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_3) * t_1)) / Float64(3.0 * fma(fma(cos(x), t_2, 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[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.43], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0005208333333333333 * N[(x * x), $MachinePrecision] + 0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] + t$95$4), $MachinePrecision]), $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$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.429999999999999993

   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. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.9

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 59.6%

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

   if -0.429999999999999993 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y + x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right)\right) \cdot \left(\color{blue}{\sin x} - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) \cdot x + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {x}^{2} + \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, {x}^{2}, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-sin.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\color{blue}{\sin x} - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       11. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       13. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       14. lift-*.f64 99.6

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift--.f64 59.4

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 59.5

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

   10. Applied rewrites 59.5%

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

Alternative 11: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := \frac{t\_0}{2} \cdot \cos y\\ \mathbf{if}\;x \leq -0.29:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + t\_4\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, x \cdot x, 0.020833333333333332\right) \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 0.5\right), t\_2, 1\right) + t\_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, t\_0\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) 1.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (* (/ t_0 2.0) (cos y))))
   (if (<= x -0.29)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) t_3) t_1))
      (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) t_4)))
     (if (<= x 5.8e-7)
       (/
        (fma
         (* (sqrt 2.0) (- (cos x) (cos y)))
         (*
          (fma
           (-
            (*
             (fma -0.0005208333333333333 (* x x) 0.010416666666666666)
             (* x x))
            0.0625)
           x
           (sin y))
          (- (sin x) (* 0.0625 (sin y))))
         2.0)
        (*
         3.0
         (+
          (fma
           (fma
            (-
             (*
              (fma -0.0006944444444444445 (* x x) 0.020833333333333332)
              (* x x))
             0.25)
            (* x x)
            0.5)
           t_2
           1.0)
          t_4)))
       (/
        (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_3) t_1))
        (* 3.0 (fma (fma (cos x) t_2 t_0) 0.5 1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = (t_0 / 2.0) * cos(y);
	double tmp;
	if (x <= -0.29) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * t_3) * t_1)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + t_4));
	} else if (x <= 5.8e-7) {
		tmp = fma((sqrt(2.0) * (cos(x) - cos(y))), (fma(((fma(-0.0005208333333333333, (x * x), 0.010416666666666666) * (x * x)) - 0.0625), x, sin(y)) * (sin(x) - (0.0625 * sin(y)))), 2.0) / (3.0 * (fma(fma(((fma(-0.0006944444444444445, (x * x), 0.020833333333333332) * (x * x)) - 0.25), (x * x), 0.5), t_2, 1.0) + t_4));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_3) * t_1)) / (3.0 * fma(fma(cos(x), t_2, t_0), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(Float64(t_0 / 2.0) * cos(y))
	tmp = 0.0
	if (x <= -0.29)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * t_3) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + t_4)));
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(fma(Float64(Float64(fma(-0.0005208333333333333, Float64(x * x), 0.010416666666666666) * Float64(x * x)) - 0.0625), x, sin(y)) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / Float64(3.0 * Float64(fma(fma(Float64(Float64(fma(-0.0006944444444444445, Float64(x * x), 0.020833333333333332) * Float64(x * x)) - 0.25), Float64(x * x), 0.5), t_2, 1.0) + t_4)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_3) * t_1)) / Float64(3.0 * fma(fma(cos(x), t_2, 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[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.29], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0005208333333333333 * N[(x * x), $MachinePrecision] + 0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(x * x), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] + t$95$4), $MachinePrecision]), $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$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.28999999999999998

   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. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.9

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 59.6%

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

   if -0.28999999999999998 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y + x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right)\right) \cdot \left(\color{blue}{\sin x} - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) \cdot x + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {x}^{2} + \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, {x}^{2}, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-sin.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\color{blue}{\sin x} - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {x}^{2}\right) - \frac{1}{4}\right), \color{blue}{\sqrt{5}} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {x}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot {x}^{2} + \frac{1}{2}, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {x}^{2}\right) - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {x}^{2}\right) - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-1}{1440} \cdot {x}^{2} + \frac{1}{48}\right) \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, {x}^{2}, \frac{1}{48}\right) \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, x \cdot x, \frac{1}{48}\right) \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, x \cdot x, \frac{1}{48}\right) \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       11. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, x \cdot x, \frac{1}{48}\right) \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, x \cdot x, \frac{1}{48}\right) \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       13. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, x \cdot x, \frac{1}{48}\right) \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       14. lift-*.f64 99.6

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, x \cdot x, 0.020833333333333332\right) \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 0.5\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, x \cdot x, 0.020833333333333332\right) \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 0.5\right), \color{blue}{\sqrt{5}} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift--.f64 59.4

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 59.5

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

   10. Applied rewrites 59.5%

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

Alternative 12: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := \frac{t\_0}{2} \cdot \cos y\\ \mathbf{if}\;x \leq -0.25:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + t\_4\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664 - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_2, 1\right) + t\_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, t\_0\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) 1.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (* (/ t_0 2.0) (cos y))))
   (if (<= x -0.25)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) t_3) t_1))
      (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) t_4)))
     (if (<= x 5.8e-7)
       (/
        (fma
         (*
          (sqrt 2.0)
          (-
           (fma (- (* (* x x) 0.041666666666666664) 0.5) (* x x) 1.0)
           (cos y)))
         (*
          (fma
           (-
            (*
             (fma -0.0005208333333333333 (* x x) 0.010416666666666666)
             (* x x))
            0.0625)
           x
           (sin y))
          (- (sin x) (* 0.0625 (sin y))))
         2.0)
        (* 3.0 (+ (fma (* 0.5 (cos x)) t_2 1.0) t_4)))
       (/
        (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_3) t_1))
        (* 3.0 (fma (fma (cos x) t_2 t_0) 0.5 1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = (t_0 / 2.0) * cos(y);
	double tmp;
	if (x <= -0.25) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * t_3) * t_1)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + t_4));
	} else if (x <= 5.8e-7) {
		tmp = fma((sqrt(2.0) * (fma((((x * x) * 0.041666666666666664) - 0.5), (x * x), 1.0) - cos(y))), (fma(((fma(-0.0005208333333333333, (x * x), 0.010416666666666666) * (x * x)) - 0.0625), x, sin(y)) * (sin(x) - (0.0625 * sin(y)))), 2.0) / (3.0 * (fma((0.5 * cos(x)), t_2, 1.0) + t_4));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_3) * t_1)) / (3.0 * fma(fma(cos(x), t_2, t_0), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(Float64(t_0 / 2.0) * cos(y))
	tmp = 0.0
	if (x <= -0.25)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * t_3) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + t_4)));
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(fma(Float64(Float64(Float64(x * x) * 0.041666666666666664) - 0.5), Float64(x * x), 1.0) - cos(y))), Float64(fma(Float64(Float64(fma(-0.0005208333333333333, Float64(x * x), 0.010416666666666666) * Float64(x * x)) - 0.0625), x, sin(y)) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / Float64(3.0 * Float64(fma(Float64(0.5 * cos(x)), t_2, 1.0) + t_4)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_3) * t_1)) / Float64(3.0 * fma(fma(cos(x), t_2, 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[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.25], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0005208333333333333 * N[(x * x), $MachinePrecision] + 0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] + t$95$4), $MachinePrecision]), $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$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.25

   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. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.9

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 59.6%

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

   if -0.25 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y + x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right)\right) \cdot \left(\color{blue}{\sin x} - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) \cdot x + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {x}^{2} + \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, {x}^{2}, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-sin.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\color{blue}{\sin x} - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24} - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. lift-*.f64 99.6

           \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664 - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664 - 0.5, x \cdot x, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift--.f64 59.4

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 59.5

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

   10. Applied rewrites 59.5%

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

Alternative 13: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := \frac{t\_0}{2} \cdot \cos y\\ \mathbf{if}\;x \leq -0.15:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + t\_4\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 0.5\right), t\_2, 1\right) + t\_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, t\_0\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) 1.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (* (/ t_0 2.0) (cos y))))
   (if (<= x -0.15)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) t_3) t_1))
      (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) t_4)))
     (if (<= x 5.8e-7)
       (/
        (fma
         (* (sqrt 2.0) (- (cos x) (cos y)))
         (*
          (fma
           (-
            (*
             (fma -0.0005208333333333333 (* x x) 0.010416666666666666)
             (* x x))
            0.0625)
           x
           (sin y))
          (- (sin x) (* 0.0625 (sin y))))
         2.0)
        (*
         3.0
         (+
          (fma
           (fma (- (* 0.020833333333333332 (* x x)) 0.25) (* x x) 0.5)
           t_2
           1.0)
          t_4)))
       (/
        (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_3) t_1))
        (* 3.0 (fma (fma (cos x) t_2 t_0) 0.5 1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = (t_0 / 2.0) * cos(y);
	double tmp;
	if (x <= -0.15) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * t_3) * t_1)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + t_4));
	} else if (x <= 5.8e-7) {
		tmp = fma((sqrt(2.0) * (cos(x) - cos(y))), (fma(((fma(-0.0005208333333333333, (x * x), 0.010416666666666666) * (x * x)) - 0.0625), x, sin(y)) * (sin(x) - (0.0625 * sin(y)))), 2.0) / (3.0 * (fma(fma(((0.020833333333333332 * (x * x)) - 0.25), (x * x), 0.5), t_2, 1.0) + t_4));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_3) * t_1)) / (3.0 * fma(fma(cos(x), t_2, t_0), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(Float64(t_0 / 2.0) * cos(y))
	tmp = 0.0
	if (x <= -0.15)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * t_3) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + t_4)));
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(fma(Float64(Float64(fma(-0.0005208333333333333, Float64(x * x), 0.010416666666666666) * Float64(x * x)) - 0.0625), x, sin(y)) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / Float64(3.0 * Float64(fma(fma(Float64(Float64(0.020833333333333332 * Float64(x * x)) - 0.25), Float64(x * x), 0.5), t_2, 1.0) + t_4)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_3) * t_1)) / Float64(3.0 * fma(fma(cos(x), t_2, 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[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.15], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0005208333333333333 * N[(x * x), $MachinePrecision] + 0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[(N[(0.020833333333333332 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] + t$95$4), $MachinePrecision]), $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$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.149999999999999994

   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. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.9

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 59.6%

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

   if -0.149999999999999994 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y + x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right)\right) \cdot \left(\color{blue}{\sin x} - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) \cdot x + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {x}^{2} + \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, {x}^{2}, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-sin.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\color{blue}{\sin x} - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{48} \cdot {x}^{2} - \frac{1}{4}\right), \color{blue}{\sqrt{5}} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{48} \cdot {x}^{2} - \frac{1}{4}\right) + \frac{1}{2}, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\left(\frac{1}{48} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2} + \frac{1}{2}, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot x, \frac{1}{2}\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. lift-*.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 0.5\right), \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 0.5\right), \color{blue}{\sqrt{5}} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift--.f64 59.4

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 59.5

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

   10. Applied rewrites 59.5%

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

Alternative 14: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := \frac{t\_0}{2} \cdot \cos y\\ \mathbf{if}\;x \leq -0.08:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + t\_4\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_2, 1\right) + t\_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, t\_0\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) 1.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (* (/ t_0 2.0) (cos y))))
   (if (<= x -0.08)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) t_3) t_1))
      (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) t_4)))
     (if (<= x 5.8e-7)
       (/
        (fma
         (* (sqrt 2.0) (- (fma (* x x) -0.5 1.0) (cos y)))
         (*
          (fma
           (-
            (*
             (fma -0.0005208333333333333 (* x x) 0.010416666666666666)
             (* x x))
            0.0625)
           x
           (sin y))
          (- (sin x) (* 0.0625 (sin y))))
         2.0)
        (* 3.0 (+ (fma (* 0.5 (cos x)) t_2 1.0) t_4)))
       (/
        (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_3) t_1))
        (* 3.0 (fma (fma (cos x) t_2 t_0) 0.5 1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = (t_0 / 2.0) * cos(y);
	double tmp;
	if (x <= -0.08) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * t_3) * t_1)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + t_4));
	} else if (x <= 5.8e-7) {
		tmp = fma((sqrt(2.0) * (fma((x * x), -0.5, 1.0) - cos(y))), (fma(((fma(-0.0005208333333333333, (x * x), 0.010416666666666666) * (x * x)) - 0.0625), x, sin(y)) * (sin(x) - (0.0625 * sin(y)))), 2.0) / (3.0 * (fma((0.5 * cos(x)), t_2, 1.0) + t_4));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_3) * t_1)) / (3.0 * fma(fma(cos(x), t_2, t_0), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(Float64(t_0 / 2.0) * cos(y))
	tmp = 0.0
	if (x <= -0.08)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * t_3) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + t_4)));
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(fma(Float64(x * x), -0.5, 1.0) - cos(y))), Float64(fma(Float64(Float64(fma(-0.0005208333333333333, Float64(x * x), 0.010416666666666666) * Float64(x * x)) - 0.0625), x, sin(y)) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / Float64(3.0 * Float64(fma(Float64(0.5 * cos(x)), t_2, 1.0) + t_4)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_3) * t_1)) / Float64(3.0 * fma(fma(cos(x), t_2, 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[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.08], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0005208333333333333 * N[(x * x), $MachinePrecision] + 0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] + t$95$4), $MachinePrecision]), $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$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0800000000000000017

   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. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.9

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 59.6%

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

   if -0.0800000000000000017 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y + x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right)\right) \cdot \left(\color{blue}{\sin x} - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}\right) \cdot x + \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {x}^{2} + \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, {x}^{2}, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot {x}^{2} - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-sin.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\color{blue}{\sin x} - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(\frac{-1}{2} \cdot {x}^{2} + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left({x}^{2} \cdot \frac{-1}{2} + 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, x \cdot x, \frac{1}{96}\right) \cdot \left(x \cdot x\right) - \frac{1}{16}, x, \sin y\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. lift-*.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, x \cdot x, 0.010416666666666666\right) \cdot \left(x \cdot x\right) - 0.0625, x, \sin y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift--.f64 59.4

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 59.5

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

   10. Applied rewrites 59.5%

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

Alternative 15: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.078:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_2\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t\_3\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, t\_0\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) 1.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- (sin y) (/ (sin x) 16.0))))
   (if (<= x -0.078)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) t_3) t_1))
      (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))
     (if (<= x 5.8e-7)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (fma (fma (* x x) -0.16666666666666666 1.0) x (* -0.0625 (sin y))))
           (fma -0.0625 x (sin y)))
          (- (cos x) (cos y))))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_2) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_2)))
       (/
        (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_3) t_1))
        (* 3.0 (fma (fma (cos x) t_2 t_0) 0.5 1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if (x <= -0.078) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * t_3) * t_1)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	} else if (x <= 5.8e-7) {
		tmp = (2.0 + (((sqrt(2.0) * fma(fma((x * x), -0.16666666666666666, 1.0), x, (-0.0625 * sin(y)))) * fma(-0.0625, x, sin(y))) * (cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, ((-0.75 * (x * x)) * t_2));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_3) * t_1)) / (3.0 * fma(fma(cos(x), t_2, t_0), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if (x <= -0.078)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * t_3) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	elseif (x <= 5.8e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(fma(Float64(x * x), -0.16666666666666666, 1.0), x, Float64(-0.0625 * sin(y)))) * fma(-0.0625, x, sin(y))) * Float64(cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_2)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_3) * t_1)) / Float64(3.0 * fma(fma(cos(x), t_2, 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[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.078], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 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], If[LessEqual[x, 5.8e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $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$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0779999999999999999

   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. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.9

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 59.6%

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

   if -0.0779999999999999999 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin y}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \frac{-1}{16} \cdot \sin \color{blue}{y}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {x}^{2}, \color{blue}{x}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {x}^{2} + 1, x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{6} + 1, x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sin.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)}\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift--.f64 59.4

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 59.5

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

   10. Applied rewrites 59.5%

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

Alternative 16: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.078:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y, \frac{t\_0}{2}, \mathsf{fma}\left(\cos x, \frac{t\_2}{2}, 1\right)\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_2\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, t\_0\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) 1.0))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.078)
     (/
      (*
       (fma (* t_1 (sqrt 2.0)) (* (pow (sin x) 2.0) -0.0625) 2.0)
       0.3333333333333333)
      (fma (cos y) (/ t_0 2.0) (fma (cos x) (/ t_2 2.0) 1.0)))
     (if (<= x 5.8e-7)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (fma (fma (* x x) -0.16666666666666666 1.0) x (* -0.0625 (sin y))))
           (fma -0.0625 x (sin y)))
          (- (cos x) (cos y))))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_2) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_2)))
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (/ (sin x) 16.0)))
          t_1))
        (* 3.0 (fma (fma (cos x) t_2 t_0) 0.5 1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.078) {
		tmp = (fma((t_1 * sqrt(2.0)), (pow(sin(x), 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), (t_0 / 2.0), fma(cos(x), (t_2 / 2.0), 1.0));
	} else if (x <= 5.8e-7) {
		tmp = (2.0 + (((sqrt(2.0) * fma(fma((x * x), -0.16666666666666666, 1.0), x, (-0.0625 * sin(y)))) * fma(-0.0625, x, sin(y))) * (cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, ((-0.75 * (x * x)) * t_2));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * t_1)) / (3.0 * fma(fma(cos(x), t_2, t_0), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.078)
		tmp = Float64(Float64(fma(Float64(t_1 * sqrt(2.0)), Float64((sin(x) ^ 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), Float64(t_0 / 2.0), fma(cos(x), Float64(t_2 / 2.0), 1.0)));
	elseif (x <= 5.8e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(fma(Float64(x * x), -0.16666666666666666, 1.0), x, Float64(-0.0625 * sin(y)))) * fma(-0.0625, x, sin(y))) * Float64(cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_2)));
	else
		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))) * t_1)) / Float64(3.0 * fma(fma(cos(x), t_2, 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[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.078], N[(N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $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] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0779999999999999999

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 59.5%

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

   if -0.0779999999999999999 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin y}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \frac{-1}{16} \cdot \sin \color{blue}{y}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {x}^{2}, \color{blue}{x}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {x}^{2} + 1, x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{6} + 1, x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sin.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)}\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift--.f64 59.4

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

   7. Applied rewrites 59.4%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 59.5

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

   10. Applied rewrites 59.5%

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

Alternative 17: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2} \cdot -0.0625\\ t_2 := \sqrt{5} - 1\\ t_3 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.078:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, t\_1, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y, \frac{t\_0}{2}, \mathsf{fma}\left(\cos x, \frac{t\_2}{2}, 1\right)\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot t\_3}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_2\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_3, t\_1, 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_2, 1\right) + \left(0.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (pow (sin x) 2.0) -0.0625))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- (cos x) (cos y))))
   (if (<= x -0.078)
     (/
      (* (fma (* (- (cos x) 1.0) (sqrt 2.0)) t_1 2.0) 0.3333333333333333)
      (fma (cos y) (/ t_0 2.0) (fma (cos x) (/ t_2 2.0) 1.0)))
     (if (<= x 5.8e-7)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (fma (fma (* x x) -0.16666666666666666 1.0) x (* -0.0625 (sin y))))
           (fma -0.0625 x (sin y)))
          t_3))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_2) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_2)))
       (/
        (fma (* (sqrt 2.0) t_3) t_1 2.0)
        (* 3.0 (+ (fma (* 0.5 (cos x)) t_2 1.0) (* (* 0.5 (cos y)) t_0))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0) * -0.0625;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.078) {
		tmp = (fma(((cos(x) - 1.0) * sqrt(2.0)), t_1, 2.0) * 0.3333333333333333) / fma(cos(y), (t_0 / 2.0), fma(cos(x), (t_2 / 2.0), 1.0));
	} else if (x <= 5.8e-7) {
		tmp = (2.0 + (((sqrt(2.0) * fma(fma((x * x), -0.16666666666666666, 1.0), x, (-0.0625 * sin(y)))) * fma(-0.0625, x, sin(y))) * t_3)) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, ((-0.75 * (x * x)) * t_2));
	} else {
		tmp = fma((sqrt(2.0) * t_3), t_1, 2.0) / (3.0 * (fma((0.5 * cos(x)), t_2, 1.0) + ((0.5 * cos(y)) * t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64((sin(x) ^ 2.0) * -0.0625)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.078)
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), t_1, 2.0) * 0.3333333333333333) / fma(cos(y), Float64(t_0 / 2.0), fma(cos(x), Float64(t_2 / 2.0), 1.0)));
	elseif (x <= 5.8e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(fma(Float64(x * x), -0.16666666666666666, 1.0), x, Float64(-0.0625 * sin(y)))) * fma(-0.0625, x, sin(y))) * t_3)) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_2)));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * t_3), t_1, 2.0) / Float64(3.0 * Float64(fma(Float64(0.5 * cos(x)), t_2, 1.0) + Float64(Float64(0.5 * cos(y)) * t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.078], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] + N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0779999999999999999

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 59.5%

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

   if -0.0779999999999999999 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin y}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \frac{-1}{16} \cdot \sin \color{blue}{y}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {x}^{2}, \color{blue}{x}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {x}^{2} + 1, x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{6} + 1, x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sin.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)}\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 99.0

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

   10. Applied rewrites 99.0%

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

   11. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 59.9

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

   13. Applied rewrites 59.9%

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

Alternative 18: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y, \frac{t\_0}{2}, \mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right)\right)}\\ \mathbf{if}\;x \leq -0.078:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (*
           (fma
            (* (- (cos x) 1.0) (sqrt 2.0))
            (* (pow (sin x) 2.0) -0.0625)
            2.0)
           0.3333333333333333)
          (fma (cos y) (/ t_0 2.0) (fma (cos x) (/ t_1 2.0) 1.0)))))
   (if (<= x -0.078)
     t_2
     (if (<= x 5.8e-7)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (fma (fma (* x x) -0.16666666666666666 1.0) x (* -0.0625 (sin y))))
           (fma -0.0625 x (sin y)))
          (- (cos x) (cos y))))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_1) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_1)))
       t_2))))

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 = (fma(((cos(x) - 1.0) * sqrt(2.0)), (pow(sin(x), 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), (t_0 / 2.0), fma(cos(x), (t_1 / 2.0), 1.0));
	double tmp;
	if (x <= -0.078) {
		tmp = t_2;
	} else if (x <= 5.8e-7) {
		tmp = (2.0 + (((sqrt(2.0) * fma(fma((x * x), -0.16666666666666666, 1.0), x, (-0.0625 * sin(y)))) * fma(-0.0625, x, sin(y))) * (cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, ((-0.75 * (x * x)) * t_1));
	} else {
		tmp = t_2;
	}
	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(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64((sin(x) ^ 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), Float64(t_0 / 2.0), fma(cos(x), Float64(t_1 / 2.0), 1.0)))
	tmp = 0.0
	if (x <= -0.078)
		tmp = t_2;
	elseif (x <= 5.8e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(fma(Float64(x * x), -0.16666666666666666, 1.0), x, Float64(-0.0625 * sin(y)))) * fma(-0.0625, x, sin(y))) * Float64(cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_1)));
	else
		tmp = t_2;
	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[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.078], t$95$2, If[LessEqual[x, 5.8e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0779999999999999999 or 5.7999999999999995e-7 < x

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 59.7%

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

   if -0.0779999999999999999 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin y}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \frac{-1}{16} \cdot \sin \color{blue}{y}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {x}^{2}, \color{blue}{x}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {x}^{2} + 1, x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{6} + 1, x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sin.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)}\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y, \frac{t\_0}{2}, \mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right)\right)}\\ \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (*
           (fma
            (* (- (cos x) 1.0) (sqrt 2.0))
            (* (pow (sin x) 2.0) -0.0625)
            2.0)
           0.3333333333333333)
          (fma (cos y) (/ t_0 2.0) (fma (cos x) (/ t_1 2.0) 1.0)))))
   (if (<= x -0.05)
     t_2
     (if (<= x 5.8e-7)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (fma -0.0625 x (sin y)))
          (- 1.0 (cos y))))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_1) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_1)))
       t_2))))

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 = (fma(((cos(x) - 1.0) * sqrt(2.0)), (pow(sin(x), 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), (t_0 / 2.0), fma(cos(x), (t_1 / 2.0), 1.0));
	double tmp;
	if (x <= -0.05) {
		tmp = t_2;
	} else if (x <= 5.8e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * fma(-0.0625, x, sin(y))) * (1.0 - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, ((-0.75 * (x * x)) * t_1));
	} else {
		tmp = t_2;
	}
	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(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64((sin(x) ^ 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), Float64(t_0 / 2.0), fma(cos(x), Float64(t_1 / 2.0), 1.0)))
	tmp = 0.0
	if (x <= -0.05)
		tmp = t_2;
	elseif (x <= 5.8e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * fma(-0.0625, x, sin(y))) * Float64(1.0 - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_1)));
	else
		tmp = t_2;
	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[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.05], t$95$2, If[LessEqual[x, 5.8e-7], 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[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.050000000000000003 or 5.7999999999999995e-7 < x

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 59.7%

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

   if -0.050000000000000003 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.5%

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

Alternative 20: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y, \frac{t\_0}{2}, \mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right)\right)}\\ \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (*
           (fma
            (* (- (cos x) 1.0) (sqrt 2.0))
            (* (pow (sin x) 2.0) -0.0625)
            2.0)
           0.3333333333333333)
          (fma (cos y) (/ t_0 2.0) (fma (cos x) (/ t_1 2.0) 1.0)))))
   (if (<= x -0.05)
     t_2
     (if (<= x 5.8e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- x (* 0.0625 (sin y)))) (fma -0.0625 x (sin y)))
          (- (cos x) (cos y))))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_1) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_1)))
       t_2))))

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 = (fma(((cos(x) - 1.0) * sqrt(2.0)), (pow(sin(x), 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), (t_0 / 2.0), fma(cos(x), (t_1 / 2.0), 1.0));
	double tmp;
	if (x <= -0.05) {
		tmp = t_2;
	} else if (x <= 5.8e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * fma(-0.0625, x, sin(y))) * (cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, ((-0.75 * (x * x)) * t_1));
	} else {
		tmp = t_2;
	}
	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(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64((sin(x) ^ 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), Float64(t_0 / 2.0), fma(cos(x), Float64(t_1 / 2.0), 1.0)))
	tmp = 0.0
	if (x <= -0.05)
		tmp = t_2;
	elseif (x <= 5.8e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(0.0625 * sin(y)))) * fma(-0.0625, x, sin(y))) * Float64(cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_1)));
	else
		tmp = t_2;
	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[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.05], t$95$2, If[LessEqual[x, 5.8e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.050000000000000003 or 5.7999999999999995e-7 < x

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 59.7%

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

   if -0.050000000000000003 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

Alternative 21: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y, \frac{t\_0}{2}, \mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right)\right)}\\ \mathbf{if}\;x \leq -0.00098:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (*
           (fma
            (* (- (cos x) 1.0) (sqrt 2.0))
            (* (pow (sin x) 2.0) -0.0625)
            2.0)
           0.3333333333333333)
          (fma (cos y) (/ t_0 2.0) (fma (cos x) (/ t_1 2.0) 1.0)))))
   (if (<= x -0.00098)
     t_2
     (if (<= x 5.8e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (* -0.0625 (sin y))) (fma -0.0625 x (sin y)))
          (- (cos x) (cos y))))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_1) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_1)))
       t_2))))

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 = (fma(((cos(x) - 1.0) * sqrt(2.0)), (pow(sin(x), 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), (t_0 / 2.0), fma(cos(x), (t_1 / 2.0), 1.0));
	double tmp;
	if (x <= -0.00098) {
		tmp = t_2;
	} else if (x <= 5.8e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (-0.0625 * sin(y))) * fma(-0.0625, x, sin(y))) * (cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, ((-0.75 * (x * x)) * t_1));
	} else {
		tmp = t_2;
	}
	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(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64((sin(x) ^ 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), Float64(t_0 / 2.0), fma(cos(x), Float64(t_1 / 2.0), 1.0)))
	tmp = 0.0
	if (x <= -0.00098)
		tmp = t_2;
	elseif (x <= 5.8e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(-0.0625 * sin(y))) * fma(-0.0625, x, sin(y))) * Float64(cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_1)));
	else
		tmp = t_2;
	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[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00098], t$95$2, If[LessEqual[x, 5.8e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.7999999999999997e-4 or 5.7999999999999995e-7 < x

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 59.7%

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

   if -9.7999999999999997e-4 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 99.2

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

   10. Applied rewrites 99.2%

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

Alternative 22: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y, \frac{t\_0}{2}, \mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right)\right)}\\ \mathbf{if}\;x \leq -0.00098:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (*
           (fma
            (* (- (cos x) 1.0) (sqrt 2.0))
            (* (pow (sin x) 2.0) -0.0625)
            2.0)
           0.3333333333333333)
          (fma (cos y) (/ t_0 2.0) (fma (cos x) (/ t_1 2.0) 1.0)))))
   (if (<= x -0.00098)
     t_2
     (if (<= x 5.8e-7)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma
         (fma 0.5 (fma t_0 (cos y) t_1) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_1)))
       t_2))))

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 = (fma(((cos(x) - 1.0) * sqrt(2.0)), (pow(sin(x), 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), (t_0 / 2.0), fma(cos(x), (t_1 / 2.0), 1.0));
	double tmp;
	if (x <= -0.00098) {
		tmp = t_2;
	} else if (x <= 5.8e-7) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, ((-0.75 * (x * x)) * t_1));
	} else {
		tmp = t_2;
	}
	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(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64((sin(x) ^ 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(cos(y), Float64(t_0 / 2.0), fma(cos(x), Float64(t_1 / 2.0), 1.0)))
	tmp = 0.0
	if (x <= -0.00098)
		tmp = t_2;
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_1)));
	else
		tmp = t_2;
	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[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00098], t$95$2, If[LessEqual[x, 5.8e-7], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.7999999999999997e-4 or 5.7999999999999995e-7 < x

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 59.7%

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

   if -9.7999999999999997e-4 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 99.2

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 99.1

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

   9. Applied rewrites 99.1%

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

Alternative 23: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right) + \left(0.5 \cdot \cos y\right) \cdot t\_1\right)}\\ \mathbf{if}\;x \leq -0.00098:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (* (- (cos x) 1.0) (sqrt 2.0))
           (* (pow (sin x) 2.0) -0.0625)
           2.0)
          (* 3.0 (+ (fma (* 0.5 (cos x)) t_0 1.0) (* (* 0.5 (cos y)) t_1))))))
   (if (<= x -0.00098)
     t_2
     (if (<= x 5.8e-7)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma
         (fma 0.5 (fma t_1 (cos y) t_0) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(((cos(x) - 1.0) * sqrt(2.0)), (pow(sin(x), 2.0) * -0.0625), 2.0) / (3.0 * (fma((0.5 * cos(x)), t_0, 1.0) + ((0.5 * cos(y)) * t_1)));
	double tmp;
	if (x <= -0.00098) {
		tmp = t_2;
	} else if (x <= 5.8e-7) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_1, cos(y), t_0), 1.0), 3.0, ((-0.75 * (x * x)) * t_0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64((sin(x) ^ 2.0) * -0.0625), 2.0) / Float64(3.0 * Float64(fma(Float64(0.5 * cos(x)), t_0, 1.0) + Float64(Float64(0.5 * cos(y)) * t_1))))
	tmp = 0.0
	if (x <= -0.00098)
		tmp = t_2;
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_1, cos(y), t_0), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] + N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00098], t$95$2, If[LessEqual[x, 5.8e-7], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.7999999999999997e-4 or 5.7999999999999995e-7 < 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. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift--.f64 99.0

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - \color{blue}{1}, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) + \color{blue}{2}}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \color{blue}{\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(\color{blue}{3} - \sqrt{5}\right)\right)} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lift--.f64 99.0

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)} \]

   10. Applied rewrites 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \color{blue}{\left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)}\right)} \]

   11. Taylor expanded in y around 0

       \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \color{blue}{2}}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \left(\frac{-1}{16} \cdot {\sin x}^{2}\right) + 2}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - 1\right), \color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16}} \cdot {\sin x}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16}} \cdot {\sin x}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       8. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       9. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin x}^{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot \color{blue}{\frac{-1}{16}}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       11. lift-sin.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       12. lift-pow.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

       13. lift-*.f64 59.7

           \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot \color{blue}{-0.0625}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   13. Applied rewrites 59.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right)}}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -9.7999999999999997e-4 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\frac{-3}{4}} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), \color{blue}{3}, \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sqrt.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos \color{blue}{y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sin y\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 99.1

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   9. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 78.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin x}^{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(t\_1 \cdot -0.0625\right) \cdot \sqrt{2}, \cos x - \cos y, 2\right)}{3}}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_2\right), 0.5, 1\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right) + t\_2 \cdot 0.5} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

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))))
   (if (<= x -0.05)
     (/
      (/ (fma (* (* t_1 -0.0625) (sqrt 2.0)) (- (cos x) (cos y)) 2.0) 3.0)
      (fma (fma t_0 (cos x) t_2) 0.5 1.0))
     (if (<= x 5.8e-7)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma
         (fma 0.5 (fma t_2 (cos y) t_0) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_0)))
       (*
        (/
         (fma (* -0.0625 t_1) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
         (+ (fma (* 0.5 (cos x)) t_0 1.0) (* t_2 0.5)))
        0.3333333333333333)))))

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 tmp;
	if (x <= -0.05) {
		tmp = (fma(((t_1 * -0.0625) * sqrt(2.0)), (cos(x) - cos(y)), 2.0) / 3.0) / fma(fma(t_0, cos(x), t_2), 0.5, 1.0);
	} else if (x <= 5.8e-7) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_2, cos(y), t_0), 1.0), 3.0, ((-0.75 * (x * x)) * t_0));
	} else {
		tmp = (fma((-0.0625 * t_1), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / (fma((0.5 * cos(x)), t_0, 1.0) + (t_2 * 0.5))) * 0.3333333333333333;
	}
	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))
	tmp = 0.0
	if (x <= -0.05)
		tmp = Float64(Float64(fma(Float64(Float64(t_1 * -0.0625) * sqrt(2.0)), Float64(cos(x) - cos(y)), 2.0) / 3.0) / fma(fma(t_0, cos(x), t_2), 0.5, 1.0));
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_2, cos(y), t_0), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_0)));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_1), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / Float64(fma(Float64(0.5 * cos(x)), t_0, 1.0) + Float64(t_2 * 0.5))) * 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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.05], N[(N[(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] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] + N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.050000000000000003

   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. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{1}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\left(\frac{1}{2} \cdot \cos x\right) \cdot \left(\sqrt{5} - 1\right) + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \color{blue}{\sqrt{5} - 1}, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \color{blue}{\sqrt{5}} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift--.f64 99.0

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - \color{blue}{1}, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{1}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2} + 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), \color{blue}{\frac{1}{2}}, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      12. lift--.f64 59.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)} \]

   7. Applied rewrites 59.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      7. lift-sqrt.f64 58.2

         \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)} \]

   10. Applied rewrites 58.2%

       \[\leadsto \frac{2 + \color{blue}{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

       3. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

   12. Applied rewrites 58.2%

       \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, \cos x - \cos y, 2\right)}{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]

   if -0.050000000000000003 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\frac{-3}{4}} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), \color{blue}{3}, \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sqrt.f64 99.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos \color{blue}{y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sin y\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 99.0

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \left(\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      11. associate-+r+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(3 - \sqrt{5}\right) \cdot 0.5} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 78.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin x}^{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{2 + \left(\left(t\_1 \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, 3\right) - \sqrt{5}, 0.5, 1\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right) + t\_2 \cdot 0.5} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

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))))
   (if (<= x -0.05)
     (/
      (+ 2.0 (* (* (* t_1 -0.0625) (sqrt 2.0)) (- (cos x) (cos y))))
      (* 3.0 (fma (- (fma t_0 (cos x) 3.0) (sqrt 5.0)) 0.5 1.0)))
     (if (<= x 5.8e-7)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma
         (fma 0.5 (fma t_2 (cos y) t_0) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_0)))
       (*
        (/
         (fma (* -0.0625 t_1) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
         (+ (fma (* 0.5 (cos x)) t_0 1.0) (* t_2 0.5)))
        0.3333333333333333)))))

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 tmp;
	if (x <= -0.05) {
		tmp = (2.0 + (((t_1 * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * fma((fma(t_0, cos(x), 3.0) - sqrt(5.0)), 0.5, 1.0));
	} else if (x <= 5.8e-7) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_2, cos(y), t_0), 1.0), 3.0, ((-0.75 * (x * x)) * t_0));
	} else {
		tmp = (fma((-0.0625 * t_1), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / (fma((0.5 * cos(x)), t_0, 1.0) + (t_2 * 0.5))) * 0.3333333333333333;
	}
	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))
	tmp = 0.0
	if (x <= -0.05)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(t_1 * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * fma(Float64(fma(t_0, cos(x), 3.0) - sqrt(5.0)), 0.5, 1.0)));
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_2, cos(y), t_0), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_0)));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_1), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / Float64(fma(Float64(0.5 * cos(x)), t_0, 1.0) + Float64(t_2 * 0.5))) * 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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.05], 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[(3.0 * N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] + N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.050000000000000003

   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. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{1}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\left(\frac{1}{2} \cdot \cos x\right) \cdot \left(\sqrt{5} - 1\right) + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \color{blue}{\sqrt{5} - 1}, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \color{blue}{\sqrt{5}} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift--.f64 99.0

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - \color{blue}{1}, 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{1}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2} + 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), \color{blue}{\frac{1}{2}}, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      12. lift--.f64 59.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)} \]

   7. Applied rewrites 59.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

      7. lift-sqrt.f64 58.2

         \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)} \]

   10. Applied rewrites 58.2%

       \[\leadsto \frac{2 + \color{blue}{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)} \]
   11. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

       2. lift-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

       3. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

       5. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

       6. associate-+r- N/A

          \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}, \frac{1}{2}, 1\right)} \]

       7. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, \frac{1}{2}, 1\right)} \]

       8. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, \frac{1}{2}, 1\right)} \]

       9. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}, \frac{1}{2}, 1\right)} \]

       10. *-commutative N/A

           \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\left(\left(\sqrt{5} - 1\right) \cdot \cos x + 3\right) - \sqrt{5}, \frac{1}{2}, 1\right)} \]

       11. lower-fma.f64 N/A

           \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, \frac{1}{2}, 1\right)} \]

       12. lift-sqrt.f64 N/A

           \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, \frac{1}{2}, 1\right)} \]

       13. lift--.f64 N/A

           \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, \frac{1}{2}, 1\right)} \]

       14. lift-cos.f64 58.2

           \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 0.5, 1\right)} \]

   12. Applied rewrites 58.2%

       \[\leadsto \frac{2 + \left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 0.5, 1\right)} \]

   if -0.050000000000000003 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\frac{-3}{4}} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), \color{blue}{3}, \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sqrt.f64 99.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos \color{blue}{y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sin y\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 99.0

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \left(\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      11. associate-+r+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 58.7%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(3 - \sqrt{5}\right) \cdot 0.5} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right) + t\_1 \cdot 0.5} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (*
          (/
           (fma
            (* -0.0625 (pow (sin x) 2.0))
            (* (- (cos x) 1.0) (sqrt 2.0))
            2.0)
           (+ (fma (* 0.5 (cos x)) t_0 1.0) (* t_1 0.5)))
          0.3333333333333333)))
   (if (<= x -0.05)
     t_2
     (if (<= x 5.8e-7)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma
         (fma 0.5 (fma t_1 (cos y) t_0) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (fma((-0.0625 * pow(sin(x), 2.0)), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / (fma((0.5 * cos(x)), t_0, 1.0) + (t_1 * 0.5))) * 0.3333333333333333;
	double tmp;
	if (x <= -0.05) {
		tmp = t_2;
	} else if (x <= 5.8e-7) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_1, cos(y), t_0), 1.0), 3.0, ((-0.75 * (x * x)) * t_0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / Float64(fma(Float64(0.5 * cos(x)), t_0, 1.0) + Float64(t_1 * 0.5))) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -0.05)
		tmp = t_2;
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_1, cos(y), t_0), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] + N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -0.05], t$95$2, If[LessEqual[x, 5.8e-7], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.050000000000000003 or 5.7999999999999995e-7 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \left(\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      11. associate-+r+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 58.5%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + \left(3 - \sqrt{5}\right) \cdot 0.5} \cdot 0.3333333333333333 \]

   if -0.050000000000000003 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\frac{-3}{4}} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), \color{blue}{3}, \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sqrt.f64 99.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos \color{blue}{y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sin y\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 99.0

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_0 \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_1, t\_3\right), 0.5, 1\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_0, t\_2, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\cos x \cdot t\_1 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.05)
     (/
      (* (fma t_2 (* t_0 -0.0625) 2.0) 0.3333333333333333)
      (fma (fma (cos x) t_1 t_3) 0.5 1.0))
     (if (<= x 5.8e-7)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma
         (fma 0.5 (fma t_3 (cos y) t_1) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_1)))
       (*
        (/
         (fma (* -0.0625 t_0) t_2 2.0)
         (fma 0.5 (+ 3.0 (- (* (cos x) t_1) (sqrt 5.0))) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (cos(x) - 1.0) * sqrt(2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.05) {
		tmp = (fma(t_2, (t_0 * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(cos(x), t_1, t_3), 0.5, 1.0);
	} else if (x <= 5.8e-7) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_3, cos(y), t_1), 1.0), 3.0, ((-0.75 * (x * x)) * t_1));
	} else {
		tmp = (fma((-0.0625 * t_0), t_2, 2.0) / fma(0.5, (3.0 + ((cos(x) * t_1) - sqrt(5.0))), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.05)
		tmp = Float64(Float64(fma(t_2, Float64(t_0 * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(cos(x), t_1, t_3), 0.5, 1.0));
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_3, cos(y), t_1), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_1)));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_0), t_2, 2.0) / fma(0.5, Float64(3.0 + Float64(Float64(cos(x) * t_1) - sqrt(5.0))), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.05], N[(N[(N[(t$95$2 * N[(t$95$0 * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$3), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$0), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(0.5 * N[(3.0 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.050000000000000003

   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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   6. Applied rewrites 58.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]

   if -0.050000000000000003 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\frac{-3}{4}} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), \color{blue}{3}, \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sqrt.f64 99.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos \color{blue}{y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sin y\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 99.0

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. associate-+r- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      9. associate--l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      15. lift--.f64 58.6

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 58.6%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_0, t\_2\right), 0.5, 1\right)}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_1, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.05)
     (/
      (* (fma t_1 (* (pow (sin x) 2.0) -0.0625) 2.0) 0.3333333333333333)
      (fma (fma (cos x) t_0 t_2) 0.5 1.0))
     (if (<= x 5.8e-7)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma
         (fma 0.5 (fma t_2 (cos y) t_0) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_0)))
       (*
        (/
         (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_1 2.0)
         (fma 0.5 (fma t_0 (cos x) t_2) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.05) {
		tmp = (fma(t_1, (pow(sin(x), 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(cos(x), t_0, t_2), 0.5, 1.0);
	} else if (x <= 5.8e-7) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_2, cos(y), t_0), 1.0), 3.0, ((-0.75 * (x * x)) * t_0));
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), t_1, 2.0) / fma(0.5, fma(t_0, cos(x), t_2), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.05)
		tmp = Float64(Float64(fma(t_1, Float64((sin(x) ^ 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(cos(x), t_0, t_2), 0.5, 1.0));
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_2, cos(y), t_0), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_0)));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), t_1, 2.0) / fma(0.5, fma(t_0, cos(x), t_2), 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[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.05], N[(N[(N[(t$95$1 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.050000000000000003

   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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   6. Applied rewrites 58.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]

   if -0.050000000000000003 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\frac{-3}{4}} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), \color{blue}{3}, \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sqrt.f64 99.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos \color{blue}{y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sin y\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 99.0

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin x \cdot \sin x\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 58.6

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 58.6%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, t\_2, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_0, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_2, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -0.05)
     (*
      (/
       (fma (* -0.0625 (pow (sin x) 2.0)) t_2 2.0)
       (fma 0.5 (- (fma (cos x) t_0 3.0) (sqrt 5.0)) 1.0))
      0.3333333333333333)
     (if (<= x 5.8e-7)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma
         (fma 0.5 (fma t_1 (cos y) t_0) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_0)))
       (*
        (/
         (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_2 2.0)
         (fma 0.5 (fma t_0 (cos x) t_1) 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 t_2 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -0.05) {
		tmp = (fma((-0.0625 * pow(sin(x), 2.0)), t_2, 2.0) / fma(0.5, (fma(cos(x), t_0, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	} else if (x <= 5.8e-7) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_1, cos(y), t_0), 1.0), 3.0, ((-0.75 * (x * x)) * t_0));
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), t_2, 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 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))
	t_2 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -0.05)
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(x) ^ 2.0)), t_2, 2.0) / fma(0.5, Float64(fma(cos(x), t_0, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333);
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_1, cos(y), t_0), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_0)));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), t_2, 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 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]}, Block[{t$95$2 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.05], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 5.8e-7], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.050000000000000003

   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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. associate-+r- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      12. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      14. lift--.f64 58.3

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 58.3%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]

   if -0.050000000000000003 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\frac{-3}{4}} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), \color{blue}{3}, \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sqrt.f64 99.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos \color{blue}{y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sin y\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 99.0

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 5.7999999999999995e-7 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin x \cdot \sin x\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 58.6

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 58.6%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 30: 78.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (*
          (/
           (fma
            (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
            (* (- (cos x) 1.0) (sqrt 2.0))
            2.0)
           (fma 0.5 (fma t_0 (cos x) t_1) 1.0))
          0.3333333333333333)))
   (if (<= x -0.05)
     t_2
     (if (<= x 5.8e-7)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma
         (fma 0.5 (fma t_1 (cos y) t_0) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -0.05) {
		tmp = t_2;
	} else if (x <= 5.8e-7) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_1, cos(y), t_0), 1.0), 3.0, ((-0.75 * (x * x)) * t_0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -0.05)
		tmp = t_2;
	elseif (x <= 5.8e-7)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, fma(t_1, cos(y), t_0), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -0.05], t$95$2, If[LessEqual[x, 5.8e-7], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.050000000000000003 or 5.7999999999999995e-7 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin x \cdot \sin x\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 58.4

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 58.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   if -0.050000000000000003 < x < 5.7999999999999995e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) \cdot 3 + \color{blue}{\frac{-3}{4}} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right), \color{blue}{3}, \frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sqrt.f64 99.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos \color{blue}{y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sin y\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-*.f64 99.0

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 60.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
    (* (- (cos x) 1.0) (sqrt 2.0))
    2.0)
   (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 1.0))
  0.3333333333333333))

C

double code(double x, double y) {
	return (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

4. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   2. lift-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   3. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin x \cdot \sin x\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   4. sqr-sin-a N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   7. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   8. lower-*.f64 60.2

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

6. Applied rewrites 60.2%

   \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
7. Add Preprocessing

Alternative 32: 43.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/ 2.0 (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 1.0))
  0.3333333333333333))

C

double code(double x, double y) {
	return (2.0 / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

4. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 43.5%

   \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 33: 41.0% accurate, 940.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

4. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 41.0%

   \[\leadsto 0.3333333333333333 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025100 
(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))))))