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

Percentage Accurate: 99.3% → 99.3%

Time: 9.9s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sin.f64 99.3

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lift-cos.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift--.f64 99.3

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

11. Applied rewrites 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sin.f64 99.3

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. distribute-lft-out N/A

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

    3. lower-fma.f64 N/A

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

11. Applied rewrites 99.3%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (*
     (* (- (sin y) (* (sin x) 0.0625)) (- (sin x) (* (sin y) 0.0625)))
     (- (cos x) (cos y)))
    (sqrt 2.0)
    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((((sin(y) - (sin(x) * 0.0625)) * (sin(x) - (sin(y) * 0.0625))) * (cos(x) - cos(y))), sqrt(2.0), 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(Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sin(x) - Float64(sin(y) * 0.0625))) * Float64(cos(x) - cos(y))), sqrt(2.0), 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[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $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%

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

Alternative 4: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \frac{\mathsf{fma}\left(t\_1, \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_2, 1\right)\right) \cdot 3}\\ \mathbf{if}\;y \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(y \cdot y, -0.010416666666666666, 0.0625\right) \cdot y\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + t\_2 \cdot \cos x\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3
         (/
          (fma
           t_1
           (* (* (- (sin x) (* 0.0625 (sin y))) (sin y)) (sqrt 2.0))
           2.0)
          (* (fma (cos y) t_0 (fma (cos x) t_2 1.0)) 3.0))))
   (if (<= y -0.2)
     t_3
     (if (<= y 1.55e-10)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (- (sin x) (* (fma (* y y) -0.010416666666666666 0.0625) y)))
           (- (sin y) (/ (sin x) 16.0)))
          t_1))
        (* 3.0 (+ (+ 1.0 (* t_2 (cos x))) (* t_0 (cos y)))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = fma(t_1, (((sin(x) - (0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0);
	double tmp;
	if (y <= -0.2) {
		tmp = t_3;
	} else if (y <= 1.55e-10) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma((y * y), -0.010416666666666666, 0.0625) * y))) * (sin(y) - (sin(x) / 16.0))) * t_1)) / (3.0 * ((1.0 + (t_2 * cos(x))) + (t_0 * cos(y))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = Float64(fma(t_1, Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -0.2)
		tmp = t_3;
	elseif (y <= 1.55e-10)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(y * y), -0.010416666666666666, 0.0625) * y))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * cos(x))) + Float64(t_0 * cos(y)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.2], t$95$3, If[LessEqual[y, 1.55e-10], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(y * y), $MachinePrecision] * -0.010416666666666666 + 0.0625), $MachinePrecision] * y), $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[(1.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.20000000000000001 or 1.55000000000000008e-10 < y

   1. Initial program 99.0%

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lift-sin.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -0.20000000000000001 < y < 1.55000000000000008e-10

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(y \cdot y, -0.010416666666666666, 0.0625\right) \cdot y\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \color{blue}{\mathsf{fma}\left(y \cdot y, -0.010416666666666666, 0.0625\right) \cdot y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\mathsf{fma}\left(t\_1, \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \mathsf{fma}\left(y \cdot y, -0.010416666666666666, 0.0625\right) \cdot y\right) \cdot \sqrt{2}\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (cos y)
           (/ (- 3.0 (sqrt 5.0)) 2.0)
           (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))
          3.0))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (/
          (fma
           t_1
           (* (* (- (sin x) (* 0.0625 (sin y))) (sin y)) (sqrt 2.0))
           2.0)
          t_0)))
   (if (<= y -0.2)
     t_2
     (if (<= y 1.55e-10)
       (/
        (fma
         t_1
         (*
          (- (sin y) (/ (sin x) 16.0))
          (*
           (- (sin x) (* (fma (* y y) -0.010416666666666666 0.0625) y))
           (sqrt 2.0)))
         2.0)
        t_0)
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = fma(t_1, (((sin(x) - (0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / t_0;
	double tmp;
	if (y <= -0.2) {
		tmp = t_2;
	} else if (y <= 1.55e-10) {
		tmp = fma(t_1, ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (fma((y * y), -0.010416666666666666, 0.0625) * y)) * sqrt(2.0))), 2.0) / t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(fma(t_1, Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / t_0)
	tmp = 0.0
	if (y <= -0.2)
		tmp = t_2;
	elseif (y <= 1.55e-10)
		tmp = Float64(fma(t_1, Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(sin(x) - Float64(fma(Float64(y * y), -0.010416666666666666, 0.0625) * y)) * sqrt(2.0))), 2.0) / t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.2], t$95$2, If[LessEqual[y, 1.55e-10], N[(N[(t$95$1 * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(y * y), $MachinePrecision] * -0.010416666666666666 + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.20000000000000001 or 1.55000000000000008e-10 < y

   1. Initial program 99.0%

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lift-sin.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -0.20000000000000001 < y < 1.55000000000000008e-10

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \mathsf{fma}\left(y \cdot y, -0.010416666666666666, 0.0625\right) \cdot y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{\mathsf{fma}\left(y \cdot y, -0.010416666666666666, 0.0625\right) \cdot y}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := 3 - \sqrt{5}\\ t_3 := \frac{\mathsf{fma}\left(t\_1, \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_2}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right) \cdot 3}\\ \mathbf{if}\;y \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - y \cdot \left(0.0625 - 0.010416666666666666 \cdot \left(y \cdot y\right)\right)\right)\right) \cdot t\_1, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (/
          (fma
           t_1
           (* (* (- (sin x) (* 0.0625 (sin y))) (sin y)) (sqrt 2.0))
           2.0)
          (* (fma (cos y) (/ t_2 2.0) (fma (cos x) (/ t_0 2.0) 1.0)) 3.0))))
   (if (<= y -0.2)
     t_3
     (if (<= y 1.55e-10)
       (/
        (fma
         (*
          (*
           (- (sin y) (* (sin x) 0.0625))
           (- (sin x) (* y (- 0.0625 (* 0.010416666666666666 (* y y))))))
          t_1)
         (sqrt 2.0)
         2.0)
        (fma (fma (* 0.5 (cos x)) t_0 1.0) 3.0 (* (* 1.5 (cos y)) t_2)))
       t_3))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(t_1, (((sin(x) - (0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / (fma(cos(y), (t_2 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0)) * 3.0);
	double tmp;
	if (y <= -0.2) {
		tmp = t_3;
	} else if (y <= 1.55e-10) {
		tmp = fma((((sin(y) - (sin(x) * 0.0625)) * (sin(x) - (y * (0.0625 - (0.010416666666666666 * (y * y)))))) * t_1), sqrt(2.0), 2.0) / fma(fma((0.5 * cos(x)), t_0, 1.0), 3.0, ((1.5 * cos(y)) * t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(fma(t_1, Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), Float64(t_2 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -0.2)
		tmp = t_3;
	elseif (y <= 1.55e-10)
		tmp = Float64(fma(Float64(Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sin(x) - Float64(y * Float64(0.0625 - Float64(0.010416666666666666 * Float64(y * y)))))) * t_1), sqrt(2.0), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_0, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_2)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.2], t$95$3, If[LessEqual[y, 1.55e-10], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 - N[(0.010416666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.20000000000000001 or 1.55000000000000008e-10 < y

   1. Initial program 99.0%

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lift-sin.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -0.20000000000000001 < y < 1.55000000000000008e-10

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - y \cdot \left(0.0625 - 0.010416666666666666 \cdot \left(y \cdot y\right)\right)\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - y \cdot \left(0.0625 - 0.010416666666666666 \cdot \left(y \cdot y\right)\right)\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_0}{2}, \mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{if}\;y \leq -0.088:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \left(1 - 0.5 \cdot \left(y \cdot y\right)\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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) (cos y))
           (* (* (- (sin x) (* 0.0625 (sin y))) (sin y)) (sqrt 2.0))
           2.0)
          (* (fma (cos y) (/ t_0 2.0) (fma (cos x) (/ t_1 2.0) 1.0)) 3.0))))
   (if (<= y -0.088)
     t_2
     (if (<= y 1.55e-10)
       (/
        (fma
         (*
          (* (- (sin y) (* (sin x) 0.0625)) (- (sin x) (* (sin y) 0.0625)))
          (- (cos x) (- 1.0 (* 0.5 (* y y)))))
         (sqrt 2.0)
         2.0)
        (fma (fma (* 0.5 (cos x)) t_1 1.0) 3.0 (* (* 1.5 (cos y)) t_0)))
       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) - cos(y)), (((sin(x) - (0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / (fma(cos(y), (t_0 / 2.0), fma(cos(x), (t_1 / 2.0), 1.0)) * 3.0);
	double tmp;
	if (y <= -0.088) {
		tmp = t_2;
	} else if (y <= 1.55e-10) {
		tmp = fma((((sin(y) - (sin(x) * 0.0625)) * (sin(x) - (sin(y) * 0.0625))) * (cos(x) - (1.0 - (0.5 * (y * y))))), sqrt(2.0), 2.0) / fma(fma((0.5 * cos(x)), t_1, 1.0), 3.0, ((1.5 * cos(y)) * t_0));
	} 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(cos(x) - cos(y)), Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), Float64(t_0 / 2.0), fma(cos(x), Float64(t_1 / 2.0), 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -0.088)
		tmp = t_2;
	elseif (y <= 1.55e-10)
		tmp = Float64(fma(Float64(Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sin(x) - Float64(sin(y) * 0.0625))) * Float64(cos(x) - Float64(1.0 - Float64(0.5 * Float64(y * y))))), sqrt(2.0), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_1, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_0)));
	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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(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] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.088], t$95$2, If[LessEqual[y, 1.55e-10], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[(1.0 - N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.087999999999999995 or 1.55000000000000008e-10 < y

   1. Initial program 99.0%

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lift-sin.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -0.087999999999999995 < y < 1.55000000000000008e-10

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.5

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

   9. Applied rewrites 99.5%

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

Alternative 8: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := 3 - \sqrt{5}\\ t_3 := \frac{\mathsf{fma}\left(t\_1, \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_2}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right) \cdot 3}\\ \mathbf{if}\;y \leq -0.018:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot t\_1, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), 3, \left(1.5 - 0.75 \cdot \left(y \cdot y\right)\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (/
          (fma
           t_1
           (* (* (- (sin x) (* 0.0625 (sin y))) (sin y)) (sqrt 2.0))
           2.0)
          (* (fma (cos y) (/ t_2 2.0) (fma (cos x) (/ t_0 2.0) 1.0)) 3.0))))
   (if (<= y -0.018)
     t_3
     (if (<= y 1.55e-10)
       (/
        (fma
         (*
          (* (- (sin y) (* (sin x) 0.0625)) (- (sin x) (* (sin y) 0.0625)))
          t_1)
         (sqrt 2.0)
         2.0)
        (fma
         (fma (* 0.5 (cos x)) t_0 1.0)
         3.0
         (* (- 1.5 (* 0.75 (* y y))) t_2)))
       t_3))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(t_1, (((sin(x) - (0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / (fma(cos(y), (t_2 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0)) * 3.0);
	double tmp;
	if (y <= -0.018) {
		tmp = t_3;
	} else if (y <= 1.55e-10) {
		tmp = fma((((sin(y) - (sin(x) * 0.0625)) * (sin(x) - (sin(y) * 0.0625))) * t_1), sqrt(2.0), 2.0) / fma(fma((0.5 * cos(x)), t_0, 1.0), 3.0, ((1.5 - (0.75 * (y * y))) * t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(fma(t_1, Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), Float64(t_2 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -0.018)
		tmp = t_3;
	elseif (y <= 1.55e-10)
		tmp = Float64(fma(Float64(Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sin(x) - Float64(sin(y) * 0.0625))) * t_1), sqrt(2.0), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_0, 1.0), 3.0, Float64(Float64(1.5 - Float64(0.75 * Float64(y * y))) * t_2)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.018], t$95$3, If[LessEqual[y, 1.55e-10], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 - N[(0.75 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0179999999999999986 or 1.55000000000000008e-10 < y

   1. Initial program 99.0%

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lift-sin.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -0.0179999999999999986 < y < 1.55000000000000008e-10

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.5

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

   9. Applied rewrites 99.5%

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

Alternative 9: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \frac{\mathsf{fma}\left(t\_1, \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_2, 1\right)\right) \cdot 3}\\ \mathbf{if}\;y \leq -0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + t\_2 \cdot \cos x\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3
         (/
          (fma
           t_1
           (* (* (- (sin x) (* 0.0625 (sin y))) (sin y)) (sqrt 2.0))
           2.0)
          (* (fma (cos y) t_0 (fma (cos x) t_2 1.0)) 3.0))))
   (if (<= y -0.05)
     t_3
     (if (<= y 1.55e-10)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (fma -0.0625 y (sin x)))
           (- (sin y) (/ (sin x) 16.0)))
          t_1))
        (* 3.0 (+ (+ 1.0 (* t_2 (cos x))) (* t_0 (cos y)))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = fma(t_1, (((sin(x) - (0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0);
	double tmp;
	if (y <= -0.05) {
		tmp = t_3;
	} else if (y <= 1.55e-10) {
		tmp = (2.0 + (((sqrt(2.0) * fma(-0.0625, y, sin(x))) * (sin(y) - (sin(x) / 16.0))) * t_1)) / (3.0 * ((1.0 + (t_2 * cos(x))) + (t_0 * cos(y))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = Float64(fma(t_1, Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -0.05)
		tmp = t_3;
	elseif (y <= 1.55e-10)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(-0.0625, y, sin(x))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * cos(x))) + Float64(t_0 * cos(y)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.05], t$95$3, If[LessEqual[y, 1.55e-10], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * y + N[Sin[x], $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[(1.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.050000000000000003 or 1.55000000000000008e-10 < y

   1. Initial program 99.0%

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lift-sin.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -0.050000000000000003 < y < 1.55000000000000008e-10

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

Alternative 10: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\mathsf{fma}\left(t\_1, \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (cos y)
           (/ (- 3.0 (sqrt 5.0)) 2.0)
           (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))
          3.0))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (/
          (fma
           t_1
           (* (* (- (sin x) (* 0.0625 (sin y))) (sin y)) (sqrt 2.0))
           2.0)
          t_0)))
   (if (<= y -0.05)
     t_2
     (if (<= y 1.55e-10)
       (/
        (fma
         t_1
         (*
          (- (sin y) (/ (sin x) 16.0))
          (* (fma -0.0625 y (sin x)) (sqrt 2.0)))
         2.0)
        t_0)
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = fma(t_1, (((sin(x) - (0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / t_0;
	double tmp;
	if (y <= -0.05) {
		tmp = t_2;
	} else if (y <= 1.55e-10) {
		tmp = fma(t_1, ((sin(y) - (sin(x) / 16.0)) * (fma(-0.0625, y, sin(x)) * sqrt(2.0))), 2.0) / t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(fma(t_1, Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / t_0)
	tmp = 0.0
	if (y <= -0.05)
		tmp = t_2;
	elseif (y <= 1.55e-10)
		tmp = Float64(fma(t_1, Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0))), 2.0) / t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.05], t$95$2, If[LessEqual[y, 1.55e-10], N[(N[(t$95$1 * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.050000000000000003 or 1.55000000000000008e-10 < y

   1. Initial program 99.0%

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lift-sin.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -0.050000000000000003 < y < 1.55000000000000008e-10

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 99.5

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

   6. Applied rewrites 99.5%

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

Alternative 11: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := 3 - \sqrt{5}\\ t_3 := \frac{\mathsf{fma}\left(t\_1, \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_2}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right) \cdot 3}\\ \mathbf{if}\;y \leq -0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - y \cdot 0.0625\right)\right) \cdot t\_1, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (/
          (fma
           t_1
           (* (* (- (sin x) (* 0.0625 (sin y))) (sin y)) (sqrt 2.0))
           2.0)
          (* (fma (cos y) (/ t_2 2.0) (fma (cos x) (/ t_0 2.0) 1.0)) 3.0))))
   (if (<= y -0.05)
     t_3
     (if (<= y 1.55e-10)
       (/
        (fma
         (* (* (- (sin y) (* (sin x) 0.0625)) (- (sin x) (* y 0.0625))) t_1)
         (sqrt 2.0)
         2.0)
        (fma (fma (* 0.5 (cos x)) t_0 1.0) 3.0 (* (* 1.5 (cos y)) t_2)))
       t_3))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(t_1, (((sin(x) - (0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / (fma(cos(y), (t_2 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0)) * 3.0);
	double tmp;
	if (y <= -0.05) {
		tmp = t_3;
	} else if (y <= 1.55e-10) {
		tmp = fma((((sin(y) - (sin(x) * 0.0625)) * (sin(x) - (y * 0.0625))) * t_1), sqrt(2.0), 2.0) / fma(fma((0.5 * cos(x)), t_0, 1.0), 3.0, ((1.5 * cos(y)) * t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(fma(t_1, Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), Float64(t_2 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -0.05)
		tmp = t_3;
	elseif (y <= 1.55e-10)
		tmp = Float64(fma(Float64(Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sin(x) - Float64(y * 0.0625))) * t_1), sqrt(2.0), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_0, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_2)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.05], t$95$3, If[LessEqual[y, 1.55e-10], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.050000000000000003 or 1.55000000000000008e-10 < y

   1. Initial program 99.0%

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lift-sin.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -0.050000000000000003 < y < 1.55000000000000008e-10

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 99.5%

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

Alternative 12: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \frac{\mathsf{fma}\left(t\_1, \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_2, 1\right)\right) \cdot 3}\\ \mathbf{if}\;y \leq -0.019:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \sqrt{2}, \mathsf{fma}\left(-0.0625 \cdot y, \sqrt{2}, \left(1.00390625 \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot y\right) \cdot t\_1}{3 \cdot \left(\left(1 + t\_2 \cdot \cos x\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3
         (/
          (fma
           t_1
           (* (* (- (sin x) (* 0.0625 (sin y))) (sin y)) (sqrt 2.0))
           2.0)
          (* (fma (cos y) t_0 (fma (cos x) t_2 1.0)) 3.0))))
   (if (<= y -0.019)
     t_3
     (if (<= y 1.55e-10)
       (/
        (+
         2.0
         (*
          (fma
           (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
           (sqrt 2.0)
           (*
            (fma
             (* -0.0625 y)
             (sqrt 2.0)
             (* (* 1.00390625 (sin x)) (sqrt 2.0)))
            y))
          t_1))
        (* 3.0 (+ (+ 1.0 (* t_2 (cos x))) (* t_0 (cos y)))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = fma(t_1, (((sin(x) - (0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0);
	double tmp;
	if (y <= -0.019) {
		tmp = t_3;
	} else if (y <= 1.55e-10) {
		tmp = (2.0 + (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), sqrt(2.0), (fma((-0.0625 * y), sqrt(2.0), ((1.00390625 * sin(x)) * sqrt(2.0))) * y)) * t_1)) / (3.0 * ((1.0 + (t_2 * cos(x))) + (t_0 * cos(y))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = Float64(fma(t_1, Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -0.019)
		tmp = t_3;
	elseif (y <= 1.55e-10)
		tmp = Float64(Float64(2.0 + Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), sqrt(2.0), Float64(fma(Float64(-0.0625 * y), sqrt(2.0), Float64(Float64(1.00390625 * sin(x)) * sqrt(2.0))) * y)) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * cos(x))) + Float64(t_0 * cos(y)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.019], t$95$3, If[LessEqual[y, 1.55e-10], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[(N[(-0.0625 * y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[(1.00390625 * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0189999999999999995 or 1.55000000000000008e-10 < y

   1. Initial program 99.0%

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lift-sin.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -0.0189999999999999995 < y < 1.55000000000000008e-10

   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

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \sqrt{2}, \mathsf{fma}\left(-0.0625 \cdot y, \sqrt{2}, \left(1.00390625 \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot y\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{\mathsf{fma}\left(\left(\sin y \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot t\_0, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_2, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_1\right)}\\ \mathbf{if}\;y \leq -0.019:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \sqrt{2}, \mathsf{fma}\left(-0.0625 \cdot y, \sqrt{2}, \left(1.00390625 \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot y\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + \frac{t\_1}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3
         (/
          (fma
           (* (* (sin y) (- (sin x) (* (sin y) 0.0625))) t_0)
           (sqrt 2.0)
           2.0)
          (fma (fma (* 0.5 (cos x)) t_2 1.0) 3.0 (* (* 1.5 (cos y)) t_1)))))
   (if (<= y -0.019)
     t_3
     (if (<= y 1.55e-10)
       (/
        (+
         2.0
         (*
          (fma
           (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
           (sqrt 2.0)
           (*
            (fma
             (* -0.0625 y)
             (sqrt 2.0)
             (* (* 1.00390625 (sin x)) (sqrt 2.0)))
            y))
          t_0))
        (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) (* (/ t_1 2.0) (cos y)))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma(((sin(y) * (sin(x) - (sin(y) * 0.0625))) * t_0), sqrt(2.0), 2.0) / fma(fma((0.5 * cos(x)), t_2, 1.0), 3.0, ((1.5 * cos(y)) * t_1));
	double tmp;
	if (y <= -0.019) {
		tmp = t_3;
	} else if (y <= 1.55e-10) {
		tmp = (2.0 + (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), sqrt(2.0), (fma((-0.0625 * y), sqrt(2.0), ((1.00390625 * sin(x)) * sqrt(2.0))) * y)) * t_0)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + ((t_1 / 2.0) * cos(y))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(fma(Float64(Float64(sin(y) * Float64(sin(x) - Float64(sin(y) * 0.0625))) * t_0), sqrt(2.0), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_2, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_1)))
	tmp = 0.0
	if (y <= -0.019)
		tmp = t_3;
	elseif (y <= 1.55e-10)
		tmp = Float64(Float64(2.0 + Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), sqrt(2.0), Float64(fma(Float64(-0.0625 * y), sqrt(2.0), Float64(Float64(1.00390625 * sin(x)) * sqrt(2.0))) * y)) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + Float64(Float64(t_1 / 2.0) * cos(y)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.019], t$95$3, If[LessEqual[y, 1.55e-10], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[(N[(-0.0625 * y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[(1.00390625 * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * t$95$0), $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$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0189999999999999995 or 1.55000000000000008e-10 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

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

      1. lift-sin.f64 64.3

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

   9. Applied rewrites 64.3%

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

   if -0.0189999999999999995 < y < 1.55000000000000008e-10

   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

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \sqrt{2}, \mathsf{fma}\left(-0.0625 \cdot y, \sqrt{2}, \left(1.00390625 \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot y\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 80.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \frac{3 - \sqrt{5}}{2}\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \mathsf{fma}\left(\cos y, t\_1, \mathsf{fma}\left(\cos x, t\_2, 1\right)\right)\\ \mathbf{if}\;x \leq -0.025:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3}}{t\_3}\\ \mathbf{elif}\;x \leq 0.039:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot \sqrt{2}, -0.0625, \mathsf{fma}\left(\sin y \cdot 1.00390625, \sqrt{2}, \left(\sqrt{2} \cdot x\right) \cdot -0.0625\right) \cdot x\right), 2\right)}{t\_3 \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \sqrt{2}\right) \cdot t\_0}{3 \cdot \left(\left(1 + t\_2 \cdot \cos x\right) + t\_1 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3 (fma (cos y) t_1 (fma (cos x) t_2 1.0))))
   (if (<= x -0.025)
     (/
      (/
       (fma
        (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
        (* (- (cos x) 1.0) (sqrt 2.0))
        2.0)
       3.0)
      t_3)
     (if (<= x 0.039)
       (/
        (fma
         t_0
         (fma
          (* (- 0.5 (* (cos (+ y y)) 0.5)) (sqrt 2.0))
          -0.0625
          (*
           (fma (* (sin y) 1.00390625) (sqrt 2.0) (* (* (sqrt 2.0) x) -0.0625))
           x))
         2.0)
        (* t_3 3.0))
       (/
        (+
         2.0
         (* (* (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) (sqrt 2.0)) t_0))
        (* 3.0 (+ (+ 1.0 (* t_2 (cos x))) (* t_1 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = (3.0 - sqrt(5.0)) / 2.0;
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = fma(cos(y), t_1, fma(cos(x), t_2, 1.0));
	double tmp;
	if (x <= -0.025) {
		tmp = (fma(((0.5 - (cos((x + x)) * 0.5)) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / 3.0) / t_3;
	} else if (x <= 0.039) {
		tmp = fma(t_0, fma(((0.5 - (cos((y + y)) * 0.5)) * sqrt(2.0)), -0.0625, (fma((sin(y) * 1.00390625), sqrt(2.0), ((sqrt(2.0) * x) * -0.0625)) * x)), 2.0) / (t_3 * 3.0);
	} else {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))) * sqrt(2.0)) * t_0)) / (3.0 * ((1.0 + (t_2 * cos(x))) + (t_1 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = fma(cos(y), t_1, fma(cos(x), t_2, 1.0))
	tmp = 0.0
	if (x <= -0.025)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / 3.0) / t_3);
	elseif (x <= 0.039)
		tmp = Float64(fma(t_0, fma(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * sqrt(2.0)), -0.0625, Float64(fma(Float64(sin(y) * 1.00390625), sqrt(2.0), Float64(Float64(sqrt(2.0) * x) * -0.0625)) * x)), 2.0) / Float64(t_3 * 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) * sqrt(2.0)) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * cos(x))) + Float64(t_1 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.025], N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[x, 0.039], N[(N[(t$95$0 * N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625 + N[(N[(N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$3 * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.025000000000000001

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 62.2%

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

   6. Applied rewrites 62.2%

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

   if -0.025000000000000001 < x < 0.0389999999999999999

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 99.4%

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

   if 0.0389999999999999999 < 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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 60.1

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

   4. Applied rewrites 60.1%

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

Alternative 15: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := \mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_2}{2}, 1\right)\right) \cdot 3\\ \mathbf{if}\;y \leq -0.019:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{t\_3}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \left(1 - 0.5 \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}, -0.0625, \mathsf{fma}\left(-0.0625 \cdot y, \sqrt{2}, \left(1.00390625 \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot y\right), 2\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot t\_0, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_2, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (* (fma (cos y) (/ t_1 2.0) (fma (cos x) (/ t_2 2.0) 1.0)) 3.0)))
   (if (<= y -0.019)
     (/
      (fma t_0 (* (* (- 0.5 (* (cos (+ y y)) 0.5)) (sqrt 2.0)) -0.0625) 2.0)
      t_3)
     (if (<= y 1.55e-10)
       (/
        (fma
         (- (cos x) (- 1.0 (* 0.5 (* y y))))
         (fma
          (* (- 0.5 (* (cos (+ x x)) 0.5)) (sqrt 2.0))
          -0.0625
          (*
           (fma (* -0.0625 y) (sqrt 2.0) (* (* 1.00390625 (sin x)) (sqrt 2.0)))
           y))
         2.0)
        t_3)
       (/
        (fma (* (* -0.0625 (pow (sin y) 2.0)) t_0) (sqrt 2.0) 2.0)
        (fma (fma (* 0.5 (cos x)) t_2 1.0) 3.0 (* (* 1.5 (cos y)) t_1)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma(cos(y), (t_1 / 2.0), fma(cos(x), (t_2 / 2.0), 1.0)) * 3.0;
	double tmp;
	if (y <= -0.019) {
		tmp = fma(t_0, (((0.5 - (cos((y + y)) * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / t_3;
	} else if (y <= 1.55e-10) {
		tmp = fma((cos(x) - (1.0 - (0.5 * (y * y)))), fma(((0.5 - (cos((x + x)) * 0.5)) * sqrt(2.0)), -0.0625, (fma((-0.0625 * y), sqrt(2.0), ((1.00390625 * sin(x)) * sqrt(2.0))) * y)), 2.0) / t_3;
	} else {
		tmp = fma(((-0.0625 * pow(sin(y), 2.0)) * t_0), sqrt(2.0), 2.0) / fma(fma((0.5 * cos(x)), t_2, 1.0), 3.0, ((1.5 * cos(y)) * t_1));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), Float64(t_2 / 2.0), 1.0)) * 3.0)
	tmp = 0.0
	if (y <= -0.019)
		tmp = Float64(fma(t_0, Float64(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / t_3);
	elseif (y <= 1.55e-10)
		tmp = Float64(fma(Float64(cos(x) - Float64(1.0 - Float64(0.5 * Float64(y * y)))), fma(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * sqrt(2.0)), -0.0625, Float64(fma(Float64(-0.0625 * y), sqrt(2.0), Float64(Float64(1.00390625 * sin(x)) * sqrt(2.0))) * y)), 2.0) / t_3);
	else
		tmp = Float64(fma(Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * t_0), sqrt(2.0), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_2, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]}, If[LessEqual[y, -0.019], N[(N[(t$95$0 * N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 1.55e-10], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[(1.0 - N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625 + N[(N[(N[(-0.0625 * y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[(1.00390625 * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0189999999999999995

   1. Initial program 99.0%

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. lift-sqrt.f64 60.8

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

   6. Applied rewrites 60.8%

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

   if -0.0189999999999999995 < y < 1.55000000000000008e-10

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + \frac{-1}{2} \cdot {y}^{2}\right)}, \mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}, \frac{-1}{16}, \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \left(\frac{257}{256} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   8. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {y}^{2}}\right), \mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}, \frac{-1}{16}, \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \left(\frac{257}{256} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 - \frac{1}{2} \cdot {\color{blue}{y}}^{2}\right), \mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}, \frac{-1}{16}, \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \left(\frac{257}{256} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 - \color{blue}{\frac{1}{2} \cdot {y}^{2}}\right), \mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}, \frac{-1}{16}, \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \left(\frac{257}{256} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 - \frac{1}{2} \cdot \color{blue}{{y}^{2}}\right), \mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}, \frac{-1}{16}, \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \left(\frac{257}{256} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 - \frac{1}{2} \cdot \left(y \cdot \color{blue}{y}\right)\right), \mathsf{fma}\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}, \frac{-1}{16}, \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \left(\frac{257}{256} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. lower-*.f64 99.4

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

   9. Applied rewrites 99.4%

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

   if 1.55000000000000008e-10 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lift-sin.f64 61.0

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

   9. Applied rewrites 61.0%

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

Alternative 16: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := t\_0 \cdot \cos y\\ t_2 := 1 - \cos y\\ t_3 := \frac{\sqrt{5} - 1}{2}\\ \mathbf{if}\;x \leq -0.01:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_3, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0057:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), t\_2 \cdot \sqrt{2}, \left(\sqrt{2} \cdot x\right) \cdot \left(\left(1.00390625 \cdot \sin y\right) \cdot t\_2\right)\right)}{3 \cdot \left(\left(1 + t\_3 \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + t\_3 \cdot \cos x\right) + t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (* t_0 (cos y)))
        (t_2 (- 1.0 (cos y)))
        (t_3 (/ (- (sqrt 5.0) 1.0) 2.0)))
   (if (<= x -0.01)
     (/
      (/
       (fma
        (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
        (* (- (cos x) 1.0) (sqrt 2.0))
        2.0)
       3.0)
      (fma (cos y) t_0 (fma (cos x) t_3 1.0)))
     (if (<= x 0.0057)
       (/
        (+
         2.0
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* t_2 (sqrt 2.0))
          (* (* (sqrt 2.0) x) (* (* 1.00390625 (sin y)) t_2))))
        (* 3.0 (+ (+ 1.0 (* t_3 (fma (* x x) -0.5 1.0))) t_1)))
       (/
        (+
         2.0
         (*
          (* (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) (sqrt 2.0))
          (- (cos x) (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_3 (cos x))) t_1)))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = t_0 * cos(y);
	double t_2 = 1.0 - cos(y);
	double t_3 = (sqrt(5.0) - 1.0) / 2.0;
	double tmp;
	if (x <= -0.01) {
		tmp = (fma(((0.5 - (cos((x + x)) * 0.5)) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / 3.0) / fma(cos(y), t_0, fma(cos(x), t_3, 1.0));
	} else if (x <= 0.0057) {
		tmp = (2.0 + fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), (t_2 * sqrt(2.0)), ((sqrt(2.0) * x) * ((1.00390625 * sin(y)) * t_2)))) / (3.0 * ((1.0 + (t_3 * fma((x * x), -0.5, 1.0))) + t_1));
	} else {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (t_3 * cos(x))) + t_1));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(t_0 * cos(y))
	t_2 = Float64(1.0 - cos(y))
	t_3 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	tmp = 0.0
	if (x <= -0.01)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / 3.0) / fma(cos(y), t_0, fma(cos(x), t_3, 1.0)));
	elseif (x <= 0.0057)
		tmp = Float64(Float64(2.0 + fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(t_2 * sqrt(2.0)), Float64(Float64(sqrt(2.0) * x) * Float64(Float64(1.00390625 * sin(y)) * t_2)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * fma(Float64(x * x), -0.5, 1.0))) + t_1)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * cos(x))) + t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.01], N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0057], N[(N[(2.0 + N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * N[(N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0100000000000000002

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 62.2%

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

   6. Applied rewrites 62.2%

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

   if -0.0100000000000000002 < x < 0.0057000000000000002

   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

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, \left(\sqrt{2} \cdot x\right) \cdot \left(\left(\frac{257}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, \left(\sqrt{2} \cdot x\right) \cdot \left(\left(\frac{257}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, \left(\sqrt{2} \cdot x\right) \cdot \left(\left(\frac{257}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left({x}^{2} \cdot \frac{-1}{2} + 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, \left(\sqrt{2} \cdot x\right) \cdot \left(\left(\frac{257}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, \left(\sqrt{2} \cdot x\right) \cdot \left(\left(\frac{257}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if 0.0057000000000000002 < 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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 60.1

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

   4. Applied rewrites 60.1%

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

Alternative 17: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{3 - \sqrt{5}}{2}\\ t_2 := t\_1 \cdot \cos y\\ t_3 := 1 - \cos y\\ t_4 := \frac{t\_0}{2}\\ \mathbf{if}\;x \leq -0.0031:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(\cos y, t\_1, \mathsf{fma}\left(\cos x, t\_4, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.00076:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), t\_3 \cdot \sqrt{2}, \left(\sqrt{2} \cdot x\right) \cdot \left(\left(1.00390625 \cdot \sin y\right) \cdot t\_3\right)\right)}{3 \cdot \left(\left(1 + 0.5 \cdot t\_0\right) + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + t\_4 \cdot \cos x\right) + t\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_2 (* t_1 (cos y)))
        (t_3 (- 1.0 (cos y)))
        (t_4 (/ t_0 2.0)))
   (if (<= x -0.0031)
     (/
      (/
       (fma
        (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
        (* (- (cos x) 1.0) (sqrt 2.0))
        2.0)
       3.0)
      (fma (cos y) t_1 (fma (cos x) t_4 1.0)))
     (if (<= x 0.00076)
       (/
        (+
         2.0
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* t_3 (sqrt 2.0))
          (* (* (sqrt 2.0) x) (* (* 1.00390625 (sin y)) t_3))))
        (* 3.0 (+ (+ 1.0 (* 0.5 t_0)) t_2)))
       (/
        (+
         2.0
         (*
          (* (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) (sqrt 2.0))
          (- (cos x) (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_4 (cos x))) 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)) / 2.0;
	double t_2 = t_1 * cos(y);
	double t_3 = 1.0 - cos(y);
	double t_4 = t_0 / 2.0;
	double tmp;
	if (x <= -0.0031) {
		tmp = (fma(((0.5 - (cos((x + x)) * 0.5)) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / 3.0) / fma(cos(y), t_1, fma(cos(x), t_4, 1.0));
	} else if (x <= 0.00076) {
		tmp = (2.0 + fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), (t_3 * sqrt(2.0)), ((sqrt(2.0) * x) * ((1.00390625 * sin(y)) * t_3)))) / (3.0 * ((1.0 + (0.5 * t_0)) + t_2));
	} else {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (t_4 * cos(x))) + t_2));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_2 = Float64(t_1 * cos(y))
	t_3 = Float64(1.0 - cos(y))
	t_4 = Float64(t_0 / 2.0)
	tmp = 0.0
	if (x <= -0.0031)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / 3.0) / fma(cos(y), t_1, fma(cos(x), t_4, 1.0)));
	elseif (x <= 0.00076)
		tmp = Float64(Float64(2.0 + fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(t_3 * sqrt(2.0)), Float64(Float64(sqrt(2.0) * x) * Float64(Float64(1.00390625 * sin(y)) * t_3)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.5 * t_0)) + t_2)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_4 * cos(x))) + t_2)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.0031], N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00076], N[(N[(2.0 + N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * N[(N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$4 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00309999999999999989

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 62.2%

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

   6. Applied rewrites 62.2%

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

   if -0.00309999999999999989 < x < 7.6000000000000004e-4

   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

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, \left(\sqrt{2} \cdot x\right) \cdot \left(\left(\frac{257}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, \left(\sqrt{2} \cdot x\right) \cdot \left(\left(\frac{257}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, \left(\sqrt{2} \cdot x\right) \cdot \left(\left(\frac{257}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift--.f64 99.3

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

   7. Applied rewrites 99.3%

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

   if 7.6000000000000004e-4 < 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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 60.2

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

   4. Applied rewrites 60.2%

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

Alternative 18: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{t\_1}{2}\\ t_3 := \frac{t\_0}{2}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(\cos y, t\_2, \mathsf{fma}\left(\cos x, t\_3, 1\right)\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_1, 3 \cdot \left(1 - -0.5 \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + t\_3 \cdot \cos x\right) + t\_2 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (/ t_1 2.0))
        (t_3 (/ t_0 2.0)))
   (if (<= x -3.4e-5)
     (/
      (/
       (fma
        (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
        (* (- (cos x) 1.0) (sqrt 2.0))
        2.0)
       3.0)
      (fma (cos y) t_2 (fma (cos x) t_3 1.0)))
     (if (<= x 2.55e-6)
       (/
        (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_1) (* 3.0 (- 1.0 (* -0.5 t_0)))))
       (/
        (+
         2.0
         (*
          (* (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) (sqrt 2.0))
          (- (cos x) (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_3 (cos x))) (* t_2 (cos y)))))))))

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 = t_1 / 2.0;
	double t_3 = t_0 / 2.0;
	double tmp;
	if (x <= -3.4e-5) {
		tmp = (fma(((0.5 - (cos((x + x)) * 0.5)) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / 3.0) / fma(cos(y), t_2, fma(cos(x), t_3, 1.0));
	} else if (x <= 2.55e-6) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_1), (3.0 * (1.0 - (-0.5 * t_0))));
	} else {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (t_3 * cos(x))) + (t_2 * cos(y))));
	}
	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(t_1 / 2.0)
	t_3 = Float64(t_0 / 2.0)
	tmp = 0.0
	if (x <= -3.4e-5)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / 3.0) / fma(cos(y), t_2, fma(cos(x), t_3, 1.0)));
	elseif (x <= 2.55e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_1), Float64(3.0 * Float64(1.0 - Float64(-0.5 * t_0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * cos(x))) + Float64(t_2 * cos(y)))));
	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[(t$95$1 / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 / 2.0), $MachinePrecision]}, If[LessEqual[x, -3.4e-5], N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * t$95$2 + N[(N[Cos[x], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e-6], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(3.0 * N[(1.0 - N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e-5

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 62.2%

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

   6. Applied rewrites 62.2%

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

   if -3.4e-5 < x < 2.5500000000000001e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.3%

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

   if 2.5500000000000001e-6 < 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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 60.3

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

   4. Applied rewrites 60.3%

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

Alternative 19: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(0.5 - t\_2 \cdot 0.5\right) \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_1, 3 \cdot \left(1 - -0.5 \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot t\_2\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))) (t_2 (cos (+ x x))))
   (if (<= x -3.4e-5)
     (/
      (/
       (fma (* (- 0.5 (* t_2 0.5)) -0.0625) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
       3.0)
      (fma (cos y) (/ t_1 2.0) (fma (cos x) (/ t_0 2.0) 1.0)))
     (if (<= x 2.55e-6)
       (/
        (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_1) (* 3.0 (- 1.0 (* -0.5 t_0)))))
       (/
        (fma
         (* (* -0.0625 (- 0.5 (* 0.5 t_2))) (- (cos x) (cos y)))
         (sqrt 2.0)
         2.0)
        (fma (fma (* 0.5 (cos x)) t_0 1.0) 3.0 (* (* 1.5 (cos y)) t_1)))))))

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 + x));
	double tmp;
	if (x <= -3.4e-5) {
		tmp = (fma(((0.5 - (t_2 * 0.5)) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / 3.0) / fma(cos(y), (t_1 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0));
	} else if (x <= 2.55e-6) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_1), (3.0 * (1.0 - (-0.5 * t_0))));
	} else {
		tmp = fma(((-0.0625 * (0.5 - (0.5 * t_2))) * (cos(x) - cos(y))), sqrt(2.0), 2.0) / fma(fma((0.5 * cos(x)), t_0, 1.0), 3.0, ((1.5 * cos(y)) * t_1));
	}
	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 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= -3.4e-5)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 - Float64(t_2 * 0.5)) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / 3.0) / fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)));
	elseif (x <= 2.55e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_1), Float64(3.0 * Float64(1.0 - Float64(-0.5 * t_0)))));
	else
		tmp = Float64(fma(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * t_2))) * Float64(cos(x) - cos(y))), sqrt(2.0), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_0, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_1)));
	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[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.4e-5], N[(N[(N[(N[(N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e-6], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(3.0 * N[(1.0 - N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e-5

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 62.2%

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

   6. Applied rewrites 62.2%

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

   if -3.4e-5 < x < 2.5500000000000001e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.3%

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

   if 2.5500000000000001e-6 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. cos-2 N/A

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

      10. cos-sum N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-+.f64 60.3

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

   9. Applied rewrites 60.3%

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

Alternative 20: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{t\_2}{2}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, t\_1, 2\right)}{3}}{\mathsf{fma}\left(\cos y, \frac{t\_0}{2}, \mathsf{fma}\left(\cos x, t\_3, 1\right)\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_0, 3 \cdot \left(1 - -0.5 \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_1, 2\right)}{3 \cdot \left(\left(1 + t\_3 \cdot \cos x\right) + \left(0.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (/ t_2 2.0)))
   (if (<= x -3.4e-5)
     (/
      (/ (fma (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625) t_1 2.0) 3.0)
      (fma (cos y) (/ t_0 2.0) (fma (cos x) t_3 1.0)))
     (if (<= x 2.55e-6)
       (/
        (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_0) (* 3.0 (- 1.0 (* -0.5 t_2)))))
       (/
        (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_1 2.0)
        (* 3.0 (+ (+ 1.0 (* t_3 (cos x))) (* (* 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) - 1.0) * sqrt(2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = t_2 / 2.0;
	double tmp;
	if (x <= -3.4e-5) {
		tmp = (fma(((0.5 - (cos((x + x)) * 0.5)) * -0.0625), t_1, 2.0) / 3.0) / fma(cos(y), (t_0 / 2.0), fma(cos(x), t_3, 1.0));
	} else if (x <= 2.55e-6) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_0), (3.0 * (1.0 - (-0.5 * t_2))));
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), t_1, 2.0) / (3.0 * ((1.0 + (t_3 * cos(x))) + ((0.5 * cos(y)) * t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(t_2 / 2.0)
	tmp = 0.0
	if (x <= -3.4e-5)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), t_1, 2.0) / 3.0) / fma(cos(y), Float64(t_0 / 2.0), fma(cos(x), t_3, 1.0)));
	elseif (x <= 2.55e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_0), Float64(3.0 * Float64(1.0 - Float64(-0.5 * t_2)))));
	else
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), t_1, 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * cos(x))) + 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[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / 2.0), $MachinePrecision]}, If[LessEqual[x, -3.4e-5], N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e-6], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(3.0 * N[(1.0 - N[(-0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(3.0 * N[(N[(1.0 + N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $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 < -3.4e-5

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 62.2%

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

   6. Applied rewrites 62.2%

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

   if -3.4e-5 < x < 2.5500000000000001e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.3%

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

   if 2.5500000000000001e-6 < 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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 60.2%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 60.2

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

   7. Applied rewrites 60.2%

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

Alternative 21: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := \frac{t\_2}{2}\\ t_4 := \frac{t\_0}{2}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, t\_1 \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(\cos y, t\_3, \mathsf{fma}\left(\cos x, t\_4, 1\right)\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_2, 3 \cdot \left(1 - -0.5 \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_1 \cdot \cosh \sinh^{-1} 1, 2\right)}{3 \cdot \left(\left(1 + t\_4 \cdot \cos x\right) + t\_3 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (/ t_2 2.0))
        (t_4 (/ t_0 2.0)))
   (if (<= x -3.4e-5)
     (/
      (/
       (fma (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625) (* t_1 (sqrt 2.0)) 2.0)
       3.0)
      (fma (cos y) t_3 (fma (cos x) t_4 1.0)))
     (if (<= x 2.55e-6)
       (/
        (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_2) (* 3.0 (- 1.0 (* -0.5 t_0)))))
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
         (* t_1 (cosh (asinh 1.0)))
         2.0)
        (* 3.0 (+ (+ 1.0 (* t_4 (cos x))) (* t_3 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = t_2 / 2.0;
	double t_4 = t_0 / 2.0;
	double tmp;
	if (x <= -3.4e-5) {
		tmp = (fma(((0.5 - (cos((x + x)) * 0.5)) * -0.0625), (t_1 * sqrt(2.0)), 2.0) / 3.0) / fma(cos(y), t_3, fma(cos(x), t_4, 1.0));
	} else if (x <= 2.55e-6) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_2), (3.0 * (1.0 - (-0.5 * t_0))));
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), (t_1 * cosh(asinh(1.0))), 2.0) / (3.0 * ((1.0 + (t_4 * cos(x))) + (t_3 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(t_2 / 2.0)
	t_4 = Float64(t_0 / 2.0)
	tmp = 0.0
	if (x <= -3.4e-5)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), Float64(t_1 * sqrt(2.0)), 2.0) / 3.0) / fma(cos(y), t_3, fma(cos(x), t_4, 1.0)));
	elseif (x <= 2.55e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_2), Float64(3.0 * Float64(1.0 - Float64(-0.5 * t_0)))));
	else
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(t_1 * cosh(asinh(1.0))), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_4 * cos(x))) + Float64(t_3 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 / 2.0), $MachinePrecision]}, If[LessEqual[x, -3.4e-5], N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * t$95$3 + N[(N[Cos[x], $MachinePrecision] * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e-6], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(3.0 * N[(1.0 - N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Cosh[N[ArcSinh[1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$4 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e-5

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 62.2%

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

   6. Applied rewrites 62.2%

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

   if -3.4e-5 < x < 2.5500000000000001e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.3%

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

   if 2.5500000000000001e-6 < 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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 60.2%

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

      1. lift-sqrt.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. cosh-asinh-rev N/A

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

      5. lower-cosh.f64 N/A

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

      6. lower-asinh.f64 60.2

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \cosh \sinh^{-1} 1, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 60.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \cosh \sinh^{-1} 1, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 79.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625\\ t_1 := 3 - \sqrt{5}\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_3 := \sqrt{5} - 1\\ t_4 := \mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_3}{2}, 1\right)\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, t\_2, 2\right)}{3}}{t\_4}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_1, 3 \cdot \left(1 - -0.5 \cdot t\_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_0, 2\right)}{t\_4 \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4 (fma (cos y) (/ t_1 2.0) (fma (cos x) (/ t_3 2.0) 1.0))))
   (if (<= x -3.4e-5)
     (/ (/ (fma t_0 t_2 2.0) 3.0) t_4)
     (if (<= x 2.55e-6)
       (/
        (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_1) (* 3.0 (- 1.0 (* -0.5 t_3)))))
       (/ (fma t_2 t_0 2.0) (* t_4 3.0))))))

C

double code(double x, double y) {
	double t_0 = (0.5 - (cos((x + x)) * 0.5)) * -0.0625;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (cos(x) - 1.0) * sqrt(2.0);
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = fma(cos(y), (t_1 / 2.0), fma(cos(x), (t_3 / 2.0), 1.0));
	double tmp;
	if (x <= -3.4e-5) {
		tmp = (fma(t_0, t_2, 2.0) / 3.0) / t_4;
	} else if (x <= 2.55e-6) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_1), (3.0 * (1.0 - (-0.5 * t_3))));
	} else {
		tmp = fma(t_2, t_0, 2.0) / (t_4 * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), Float64(t_3 / 2.0), 1.0))
	tmp = 0.0
	if (x <= -3.4e-5)
		tmp = Float64(Float64(fma(t_0, t_2, 2.0) / 3.0) / t_4);
	elseif (x <= 2.55e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_1), Float64(3.0 * Float64(1.0 - Float64(-0.5 * t_3)))));
	else
		tmp = Float64(fma(t_2, t_0, 2.0) / Float64(t_4 * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $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]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$3 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-5], N[(N[(N[(t$95$0 * t$95$2 + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[x, 2.55e-6], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(3.0 * N[(1.0 - N[(-0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * t$95$0 + 2.0), $MachinePrecision] / N[(t$95$4 * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e-5

   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

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\sqrt{2} \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 62.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}} \]

   if -3.4e-5 < x < 2.5500000000000001e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   9. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 3 \cdot \left(1 - -0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   if 2.5500000000000001e-6 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. sqr-sin-a-rev N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\cos x - 1\right)\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 60.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 79.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot t\_3\right) \cdot t\_0\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_2, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_1\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_1, 3 \cdot \left(1 - -0.5 \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, \left(0.5 - t\_3 \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_2}{2}, 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (cos (+ x x))))
   (if (<= x -3.4e-5)
     (/
      (fma (* -0.0625 (* (- 0.5 (* 0.5 t_3)) t_0)) (sqrt 2.0) 2.0)
      (fma (fma (* 0.5 (cos x)) t_2 1.0) 3.0 (* (* 1.5 (cos y)) t_1)))
     (if (<= x 2.55e-6)
       (/
        (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_1) (* 3.0 (- 1.0 (* -0.5 t_2)))))
       (/
        (fma (* t_0 (sqrt 2.0)) (* (- 0.5 (* t_3 0.5)) -0.0625) 2.0)
        (* (fma (cos y) (/ t_1 2.0) (fma (cos x) (/ t_2 2.0) 1.0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = cos((x + x));
	double tmp;
	if (x <= -3.4e-5) {
		tmp = fma((-0.0625 * ((0.5 - (0.5 * t_3)) * t_0)), sqrt(2.0), 2.0) / fma(fma((0.5 * cos(x)), t_2, 1.0), 3.0, ((1.5 * cos(y)) * t_1));
	} else if (x <= 2.55e-6) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_1), (3.0 * (1.0 - (-0.5 * t_2))));
	} else {
		tmp = fma((t_0 * sqrt(2.0)), ((0.5 - (t_3 * 0.5)) * -0.0625), 2.0) / (fma(cos(y), (t_1 / 2.0), fma(cos(x), (t_2 / 2.0), 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= -3.4e-5)
		tmp = Float64(fma(Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * t_3)) * t_0)), sqrt(2.0), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_2, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_1)));
	elseif (x <= 2.55e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_1), Float64(3.0 * Float64(1.0 - Float64(-0.5 * t_2)))));
	else
		tmp = Float64(fma(Float64(t_0 * sqrt(2.0)), Float64(Float64(0.5 - Float64(t_3 * 0.5)) * -0.0625), 2.0) / Float64(fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), Float64(t_2 / 2.0), 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.4e-5], N[(N[(N[(-0.0625 * N[(N[(0.5 - N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e-6], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(3.0 * N[(1.0 - N[(-0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e-5

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   3. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \cos x}, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \color{blue}{\cos x}, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. sqr-sin-a-rev N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(-2 \cdot x\right)\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(-2 \cdot x\right)\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. cos-2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos x \cdot \cos x - \sin x \cdot \sin x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. cos-sum N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      15. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      16. lift--.f64 62.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Applied rewrites 62.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.5 \cdot \cos x}, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -3.4e-5 < x < 2.5500000000000001e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   9. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 3 \cdot \left(1 - -0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   if 2.5500000000000001e-6 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. sqr-sin-a-rev N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\left(\sqrt{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\cos x - 1\right)\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 60.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 79.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_0, 3 \cdot \left(1 - -0.5 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (fma
           (* -0.0625 (* (- 0.5 (* 0.5 (cos (+ x x)))) (- (cos x) 1.0)))
           (sqrt 2.0)
           2.0)
          (fma (fma (* 0.5 (cos x)) t_1 1.0) 3.0 (* (* 1.5 (cos y)) t_0)))))
   (if (<= x -3.4e-5)
     t_2
     (if (<= x 2.55e-6)
       (/
        (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_0) (* 3.0 (- 1.0 (* -0.5 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((-0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (cos(x) - 1.0))), sqrt(2.0), 2.0) / fma(fma((0.5 * cos(x)), t_1, 1.0), 3.0, ((1.5 * cos(y)) * t_0));
	double tmp;
	if (x <= -3.4e-5) {
		tmp = t_2;
	} else if (x <= 2.55e-6) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_0), (3.0 * (1.0 - (-0.5 * 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(fma(Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(cos(x) - 1.0))), sqrt(2.0), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_1, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_0)))
	tmp = 0.0
	if (x <= -3.4e-5)
		tmp = t_2;
	elseif (x <= 2.55e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_0), Float64(3.0 * Float64(1.0 - Float64(-0.5 * 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[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-5], t$95$2, If[LessEqual[x, 2.55e-6], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(3.0 * N[(1.0 - N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4e-5 or 2.5500000000000001e-6 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   3. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \cos x}, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \color{blue}{\cos x}, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. sqr-sin-a-rev N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(-2 \cdot x\right)\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(-2 \cdot x\right)\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. cos-2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos x \cdot \cos x - \sin x \cdot \sin x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. cos-sum N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      15. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      16. lift--.f64 61.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Applied rewrites 61.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\cos x - 1\right)\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.5 \cdot \cos x}, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -3.4e-5 < x < 2.5500000000000001e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   9. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 3 \cdot \left(1 - -0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_0, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right) + 0.5 \cdot t\_2} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_2, 3 \cdot \left(1 - -0.5 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, t\_0 \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_2\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -8.5e-5)
     (*
      (/
       (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_0 2.0)
       (+ (fma (* 0.5 (cos x)) t_1 1.0) (* 0.5 t_2)))
      0.3333333333333333)
     (if (<= x 2.55e-6)
       (/
        (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_2) (* 3.0 (- 1.0 (* -0.5 t_1)))))
       (*
        (/
         (fma -0.0625 (* t_0 (- 0.5 (* (cos (+ x x)) 0.5))) 2.0)
         (fma 0.5 (fma t_1 (cos x) t_2) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = (cos(x) - 1.0) * sqrt(2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -8.5e-5) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), t_0, 2.0) / (fma((0.5 * cos(x)), t_1, 1.0) + (0.5 * t_2))) * 0.3333333333333333;
	} else if (x <= 2.55e-6) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_2), (3.0 * (1.0 - (-0.5 * t_1))));
	} else {
		tmp = (fma(-0.0625, (t_0 * (0.5 - (cos((x + x)) * 0.5))), 2.0) / fma(0.5, fma(t_1, cos(x), t_2), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -8.5e-5)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), t_0, 2.0) / Float64(fma(Float64(0.5 * cos(x)), t_1, 1.0) + Float64(0.5 * t_2))) * 0.3333333333333333);
	elseif (x <= 2.55e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_2), Float64(3.0 * Float64(1.0 - Float64(-0.5 * t_1)))));
	else
		tmp = Float64(Float64(fma(-0.0625, Float64(t_0 * Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5))), 2.0) / fma(0.5, fma(t_1, cos(x), t_2), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e-5], 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$0 + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 2.55e-6], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(3.0 * N[(1.0 - N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[(t$95$0 * N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * 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 < -8.500000000000001e-5

   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 61.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      11. associate-+r+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 61.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)} \cdot 0.3333333333333333 \]

   if -8.500000000000001e-5 < x < 2.5500000000000001e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   9. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 3 \cdot \left(1 - -0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   if 2.5500000000000001e-6 < 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 59.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 79.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_0, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right) + 0.5 \cdot t\_2} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_2, \mathsf{fma}\left(0.5, t\_1, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, t\_0 \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_2\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -8.5e-5)
     (*
      (/
       (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_0 2.0)
       (+ (fma (* 0.5 (cos x)) t_1 1.0) (* 0.5 t_2)))
      0.3333333333333333)
     (if (<= x 2.55e-6)
       (/
        (fma
         (* (- 1.0 (cos y)) (sqrt 2.0))
         (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
         2.0)
        (fma (* 1.5 (cos y)) t_2 (* (fma 0.5 t_1 1.0) 3.0)))
       (*
        (/
         (fma -0.0625 (* t_0 (- 0.5 (* (cos (+ x x)) 0.5))) 2.0)
         (fma 0.5 (fma t_1 (cos x) t_2) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = (cos(x) - 1.0) * sqrt(2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -8.5e-5) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), t_0, 2.0) / (fma((0.5 * cos(x)), t_1, 1.0) + (0.5 * t_2))) * 0.3333333333333333;
	} else if (x <= 2.55e-6) {
		tmp = fma(((1.0 - cos(y)) * sqrt(2.0)), ((0.5 - (cos((y + y)) * 0.5)) * -0.0625), 2.0) / fma((1.5 * cos(y)), t_2, (fma(0.5, t_1, 1.0) * 3.0));
	} else {
		tmp = (fma(-0.0625, (t_0 * (0.5 - (cos((x + x)) * 0.5))), 2.0) / fma(0.5, fma(t_1, cos(x), t_2), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -8.5e-5)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), t_0, 2.0) / Float64(fma(Float64(0.5 * cos(x)), t_1, 1.0) + Float64(0.5 * t_2))) * 0.3333333333333333);
	elseif (x <= 2.55e-6)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), 2.0) / fma(Float64(1.5 * cos(y)), t_2, Float64(fma(0.5, t_1, 1.0) * 3.0)));
	else
		tmp = Float64(Float64(fma(-0.0625, Float64(t_0 * Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5))), 2.0) / fma(0.5, fma(t_1, cos(x), t_2), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e-5], 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$0 + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 2.55e-6], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[(0.5 * t$95$1 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[(t$95$0 * N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * 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 < -8.500000000000001e-5

   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 61.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      9. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      11. associate-+r+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 61.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)} \cdot 0.3333333333333333 \]

   if -8.500000000000001e-5 < x < 2.5500000000000001e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   6. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right) \cdot 3\right)}} \]

   if 2.5500000000000001e-6 < 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 59.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 79.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \cos \left(x + x\right) \cdot 0.5\\ t_1 := 3 - \sqrt{5}\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_3 := \sqrt{5} - 1\\ t_4 := \mathsf{fma}\left(t\_3, \cos x, t\_1\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_0 \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(t\_4, 0.5, 1\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_1, \mathsf{fma}\left(0.5, t\_3, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, t\_2 \cdot t\_0, 2\right)}{\mathsf{fma}\left(0.5, t\_4, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.5 (* (cos (+ x x)) 0.5)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4 (fma t_3 (cos x) t_1)))
   (if (<= x -8.5e-5)
     (/ (* (fma t_2 (* t_0 -0.0625) 2.0) 0.3333333333333333) (fma t_4 0.5 1.0))
     (if (<= x 2.55e-6)
       (/
        (fma
         (* (- 1.0 (cos y)) (sqrt 2.0))
         (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
         2.0)
        (fma (* 1.5 (cos y)) t_1 (* (fma 0.5 t_3 1.0) 3.0)))
       (*
        (/ (fma -0.0625 (* t_2 t_0) 2.0) (fma 0.5 t_4 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = 0.5 - (cos((x + x)) * 0.5);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (cos(x) - 1.0) * sqrt(2.0);
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = fma(t_3, cos(x), t_1);
	double tmp;
	if (x <= -8.5e-5) {
		tmp = (fma(t_2, (t_0 * -0.0625), 2.0) * 0.3333333333333333) / fma(t_4, 0.5, 1.0);
	} else if (x <= 2.55e-6) {
		tmp = fma(((1.0 - cos(y)) * sqrt(2.0)), ((0.5 - (cos((y + y)) * 0.5)) * -0.0625), 2.0) / fma((1.5 * cos(y)), t_1, (fma(0.5, t_3, 1.0) * 3.0));
	} else {
		tmp = (fma(-0.0625, (t_2 * t_0), 2.0) / fma(0.5, t_4, 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = fma(t_3, cos(x), t_1)
	tmp = 0.0
	if (x <= -8.5e-5)
		tmp = Float64(Float64(fma(t_2, Float64(t_0 * -0.0625), 2.0) * 0.3333333333333333) / fma(t_4, 0.5, 1.0));
	elseif (x <= 2.55e-6)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), 2.0) / fma(Float64(1.5 * cos(y)), t_1, Float64(fma(0.5, t_3, 1.0) * 3.0)));
	else
		tmp = Float64(Float64(fma(-0.0625, Float64(t_2 * t_0), 2.0) / fma(0.5, t_4, 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $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]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[x, -8.5e-5], N[(N[(N[(t$95$2 * N[(t$95$0 * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(t$95$4 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e-6], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(0.5 * t$95$3 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[(t$95$2 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.500000000000001e-5

   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 61.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   6. Applied rewrites 61.1%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]

   if -8.500000000000001e-5 < x < 2.5500000000000001e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   6. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right) \cdot 3\right)}} \]

   if 2.5500000000000001e-6 < 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 59.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 79.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \cos \left(x + x\right) \cdot 0.5\\ t_1 := 3 - \sqrt{5}\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_3 := \sqrt{5} - 1\\ t_4 := \mathsf{fma}\left(t\_3, \cos x, t\_1\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_0 \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(t\_4, 0.5, 1\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_3\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, t\_2 \cdot t\_0, 2\right)}{\mathsf{fma}\left(0.5, t\_4, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.5 (* (cos (+ x x)) 0.5)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4 (fma t_3 (cos x) t_1)))
   (if (<= x -8.5e-5)
     (/ (* (fma t_2 (* t_0 -0.0625) 2.0) 0.3333333333333333) (fma t_4 0.5 1.0))
     (if (<= x 2.55e-6)
       (*
        (/
         (fma
          (* (- 1.0 (cos y)) (sqrt 2.0))
          (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
          2.0)
         (fma 0.5 (fma t_1 (cos y) t_3) 1.0))
        0.3333333333333333)
       (*
        (/ (fma -0.0625 (* t_2 t_0) 2.0) (fma 0.5 t_4 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = 0.5 - (cos((x + x)) * 0.5);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (cos(x) - 1.0) * sqrt(2.0);
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = fma(t_3, cos(x), t_1);
	double tmp;
	if (x <= -8.5e-5) {
		tmp = (fma(t_2, (t_0 * -0.0625), 2.0) * 0.3333333333333333) / fma(t_4, 0.5, 1.0);
	} else if (x <= 2.55e-6) {
		tmp = (fma(((1.0 - cos(y)) * sqrt(2.0)), ((0.5 - (cos((y + y)) * 0.5)) * -0.0625), 2.0) / fma(0.5, fma(t_1, cos(y), t_3), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (fma(-0.0625, (t_2 * t_0), 2.0) / fma(0.5, t_4, 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = fma(t_3, cos(x), t_1)
	tmp = 0.0
	if (x <= -8.5e-5)
		tmp = Float64(Float64(fma(t_2, Float64(t_0 * -0.0625), 2.0) * 0.3333333333333333) / fma(t_4, 0.5, 1.0));
	elseif (x <= 2.55e-6)
		tmp = Float64(Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), 2.0) / fma(0.5, fma(t_1, cos(y), t_3), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(-0.0625, Float64(t_2 * t_0), 2.0) / fma(0.5, t_4, 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $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]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[x, -8.5e-5], N[(N[(N[(t$95$2 * N[(t$95$0 * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(t$95$4 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e-6], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(-0.0625 * N[(t$95$2 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.500000000000001e-5

   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 61.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   6. Applied rewrites 61.1%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]

   if -8.500000000000001e-5 < x < 2.5500000000000001e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333} \]

   if 2.5500000000000001e-6 < 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 59.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 60.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (*
   (fma
    (* (- (cos x) 1.0) (sqrt 2.0))
    (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
    2.0)
   0.3333333333333333)
  (fma (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 0.5 1.0)))

C

double code(double x, double y) {
	return (fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 0.5, 1.0);
}

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 0.5, 1.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]

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 60.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

6. Applied rewrites 60.5%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]
7. Add Preprocessing

Alternative 30: 60.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    -0.0625
    (* (* (- (cos x) 1.0) (sqrt 2.0)) (- 0.5 (* (cos (+ x x)) 0.5)))
    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, (((cos(x) - 1.0) * sqrt(2.0)) * (0.5 - (cos((x + x)) * 0.5))), 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(-0.0625, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5))), 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[(-0.0625 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $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 60.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

6. Applied rewrites 60.5%

   \[\leadsto \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
7. Add Preprocessing

Alternative 31: 45.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   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 / (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 / (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 / (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 / (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(2.0 / 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 / (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[(2.0 / 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]

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

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\sqrt{2} \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

4. Applied rewrites 62.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

7. Applied rewrites 45.3%

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
8. Add Preprocessing

Alternative 32: 42.9% accurate, 5.5× 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 60.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

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 42.9%

   \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 33: 40.4% accurate, 316.7× 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 60.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 40.4%

   \[\leadsto 0.3333333333333333 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(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))))))