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

Percentage Accurate: 99.3% → 99.3%

Time: 9.9s

Alternatives: 29

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(1, \cos x, \cos \left(\pi \cdot -0.5\right) \cdot \sin x\right)\\ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(t\_0 - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), t\_0, 1\right) \cdot 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 1.0 (cos x) (* (cos (* PI -0.5)) (sin x)))))
   (/
    (+
     2.0
     (*
      (*
       (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
       (- (sin y) (/ (sin x) 16.0)))
      (- t_0 (cos y))))
    (fma
     (* 3.0 (cos y))
     0.38196601125010515
     (* (fma (fma (sqrt 5.0) 0.5 -0.5) t_0 1.0) 3.0)))))

C

double code(double x, double y) {
	double t_0 = fma(1.0, cos(x), (cos((((double) M_PI) * -0.5)) * sin(x)));
	return (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (t_0 - cos(y)))) / fma((3.0 * cos(y)), 0.38196601125010515, (fma(fma(sqrt(5.0), 0.5, -0.5), t_0, 1.0) * 3.0));
}

Julia

function code(x, y)
	t_0 = fma(1.0, cos(x), Float64(cos(Float64(pi * -0.5)) * sin(x)))
	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(t_0 - cos(y)))) / fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), t_0, 1.0) * 3.0)))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 * N[Cos[x], $MachinePrecision] + N[(N[Cos[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.3%

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

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

   4. distribute-lft-in N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

3. Applied rewrites 99.3%

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

4. Evaluated real constant 99.3%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\cos x} - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\sin \left(x + \frac{\mathsf{PI}\left(\right)}{2}\right)} - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   3. lift-PI.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sin \left(x + \frac{\color{blue}{\pi}}{2}\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   4. mult-flip-rev N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sin \left(x + \color{blue}{\pi \cdot \frac{1}{2}}\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sin \left(x + \pi \cdot \color{blue}{\frac{1}{2}}\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sin \left(x + \color{blue}{\pi \cdot \frac{1}{2}}\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sin \color{blue}{\left(\pi \cdot \frac{1}{2} + x\right)} - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   8. sin-sum N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos x + \cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin x\right)} - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin x\right)} - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   10. lower-sin.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\sin \left(\pi \cdot \frac{1}{2}\right)}, \cos x, \cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \color{blue}{\cos x}, \cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin x}\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \color{blue}{\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin x}\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   14. cos-neg-rev N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{1}{2}\right)\right)} \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   15. lower-cos.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{1}{2}\right)\right)} \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\mathsf{neg}\left(\color{blue}{\pi \cdot \frac{1}{2}}\right)\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   17. distribute-rgt-neg-in N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \color{blue}{\left(\pi \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \color{blue}{\frac{-1}{2}}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   19. lower-*.f64 99.3

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot 0.5\right), \cos x, \cos \color{blue}{\left(\pi \cdot -0.5\right)} \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

6. Applied rewrites 99.3%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\sin \left(\pi \cdot 0.5\right), \cos x, \cos \left(\pi \cdot -0.5\right) \cdot \sin x\right)} - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x}, 1\right) \cdot 3\right)} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\sin \left(x + \frac{\mathsf{PI}\left(\right)}{2}\right)}, 1\right) \cdot 3\right)} \]

   3. lift-PI.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \sin \left(x + \frac{\color{blue}{\pi}}{2}\right), 1\right) \cdot 3\right)} \]

   4. mult-flip-rev N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \sin \left(x + \color{blue}{\pi \cdot \frac{1}{2}}\right), 1\right) \cdot 3\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \sin \left(x + \pi \cdot \color{blue}{\frac{1}{2}}\right), 1\right) \cdot 3\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \sin \left(x + \color{blue}{\pi \cdot \frac{1}{2}}\right), 1\right) \cdot 3\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \sin \color{blue}{\left(\pi \cdot \frac{1}{2} + x\right)}, 1\right) \cdot 3\right)} \]

   8. sin-sum N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos x + \cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin x}, 1\right) \cdot 3\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin x\right)}, 1\right) \cdot 3\right)} \]

   10. lower-sin.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\color{blue}{\sin \left(\pi \cdot \frac{1}{2}\right)}, \cos x, \cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin x\right), 1\right) \cdot 3\right)} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \color{blue}{\cos x}, \cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin x\right), 1\right) \cdot 3\right)} \]

   12. lift-sin.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin x}\right), 1\right) \cdot 3\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \color{blue}{\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin x}\right), 1\right) \cdot 3\right)} \]

   14. cos-neg-rev N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{1}{2}\right)\right)} \cdot \sin x\right), 1\right) \cdot 3\right)} \]

   15. lower-cos.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{1}{2}\right)\right)} \cdot \sin x\right), 1\right) \cdot 3\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\mathsf{neg}\left(\color{blue}{\pi \cdot \frac{1}{2}}\right)\right) \cdot \sin x\right), 1\right) \cdot 3\right)} \]

   17. distribute-rgt-neg-in N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \color{blue}{\left(\pi \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \sin x\right), 1\right) \cdot 3\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \color{blue}{\frac{-1}{2}}\right) \cdot \sin x\right), 1\right) \cdot 3\right)} \]

   19. lower-*.f64 99.3

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot 0.5\right), \cos x, \cos \left(\pi \cdot -0.5\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\sin \left(\pi \cdot 0.5\right), \cos x, \cos \color{blue}{\left(\pi \cdot -0.5\right)} \cdot \sin x\right), 1\right) \cdot 3\right)} \]

8. Applied rewrites 99.3%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\sin \left(\pi \cdot 0.5\right), \cos x, \cos \left(\pi \cdot -0.5\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \color{blue}{\mathsf{fma}\left(\sin \left(\pi \cdot 0.5\right), \cos x, \cos \left(\pi \cdot -0.5\right) \cdot \sin x\right)}, 1\right) \cdot 3\right)} \]

9. Evaluated real constant 99.3%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{1}, \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\sin \left(\pi \cdot \frac{1}{2}\right), \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right), 1\right) \cdot 3\right)} \]

10. Evaluated real constant 99.3%

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(1, \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right) - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\color{blue}{1}, \cos x, \cos \left(\pi \cdot \frac{-1}{2}\right) \cdot \sin x\right), 1\right) \cdot 3\right)} \]
11. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(0.6180339887498949, \cos x, 1\right) \cdot 3\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (fma
   (* 3.0 (cos y))
   0.38196601125010515
   (* (fma 0.6180339887498949 (cos x) 1.0) 3.0))))

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)))) / fma((3.0 * cos(y)), 0.38196601125010515, (fma(0.6180339887498949, cos(x), 1.0) * 3.0));
}

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)))) / fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(fma(0.6180339887498949, cos(x), 1.0) * 3.0)))
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[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(N[(0.6180339887498949 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

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

   4. distribute-lft-in N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

3. Applied rewrites 99.3%

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

4. Evaluated real constant 99.3%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

5. Evaluated real constant 99.3%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\color{blue}{\frac{347922205179541}{562949953421312}}, \cos x, 1\right) \cdot 3\right)} \]
6. Add Preprocessing

Alternative 3: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 \cdot \cos y\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\\ t_3 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0}{\mathsf{fma}\left(t\_1, 0.5 \cdot \left(3 - \sqrt{5}\right), t\_2\right)}\\ \mathbf{if}\;y \leq -0.08:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 4:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - y \cdot \left(0.0625 + -0.010416666666666666 \cdot {y}^{2}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(t\_1, 0.38196601125010515, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* 3.0 (cos y)))
        (t_2 (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0))
        (t_3
         (/
          (+
           2.0
           (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0))
          (fma t_1 (* 0.5 (- 3.0 (sqrt 5.0))) t_2))))
   (if (<= y -0.08)
     t_3
     (if (<= y 4.0)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (- (sin x) (* y (+ 0.0625 (* -0.010416666666666666 (pow y 2.0))))))
           (- (sin y) (/ (sin x) 16.0)))
          t_0))
        (fma t_1 0.38196601125010515 t_2))
       t_3))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 * cos(y);
	double t_2 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0;
	double t_3 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / fma(t_1, (0.5 * (3.0 - sqrt(5.0))), t_2);
	double tmp;
	if (y <= -0.08) {
		tmp = t_3;
	} else if (y <= 4.0) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y * (0.0625 + (-0.010416666666666666 * pow(y, 2.0)))))) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma(t_1, 0.38196601125010515, t_2);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 * cos(y))
	t_2 = Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0)
	t_3 = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0)) / fma(t_1, Float64(0.5 * Float64(3.0 - sqrt(5.0))), t_2))
	tmp = 0.0
	if (y <= -0.08)
		tmp = t_3;
	elseif (y <= 4.0)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y * Float64(0.0625 + Float64(-0.010416666666666666 * (y ^ 2.0)))))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(t_1, 0.38196601125010515, t_2));
	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[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$3 = 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[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.08], t$95$3, If[LessEqual[y, 4.0], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 + N[(-0.010416666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * 0.38196601125010515 + t$95$2), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0800000000000000017 or 4 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   6. Applied rewrites 64.5%

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

   if -0.0800000000000000017 < y < 4

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \color{blue}{y \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot {y}^{2}\right)}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - y \cdot \color{blue}{\left(\frac{1}{16} + \frac{-1}{96} \cdot {y}^{2}\right)}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - y \cdot \left(\frac{1}{16} + \color{blue}{\frac{-1}{96} \cdot {y}^{2}}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - y \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot \color{blue}{{y}^{2}}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-pow.f64 51.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - y \cdot \left(0.0625 + -0.010416666666666666 \cdot {y}^{\color{blue}{2}}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 51.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \color{blue}{y \cdot \left(0.0625 + -0.010416666666666666 \cdot {y}^{2}\right)}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\\ t_1 := \cos x - \cos y\\ t_2 := 2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1\\ t_3 := \mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)\\ \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{t\_2}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_0 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_0}{16}\right)\right) \cdot t\_1}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_3}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0)))))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (+
          2.0
          (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_1)))
        (t_3
         (fma
          (* 3.0 (cos y))
          0.38196601125010515
          (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0))))
   (if (<= x -3.8)
     (/
      t_2
      (*
       3.0
       (+
        (+ 1.0 (* 0.6180339887498949 (cos x)))
        (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
     (if (<= x 0.55)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_0 (/ (sin y) 16.0))) (- (sin y) (/ t_0 16.0)))
          t_1))
        t_3)
       (/ t_2 t_3)))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)));
	double t_1 = cos(x) - cos(y);
	double t_2 = 2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_1);
	double t_3 = fma((3.0 * cos(y)), 0.38196601125010515, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	double tmp;
	if (x <= -3.8) {
		tmp = t_2 / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	} else if (x <= 0.55) {
		tmp = (2.0 + (((sqrt(2.0) * (t_0 - (sin(y) / 16.0))) * (sin(y) - (t_0 / 16.0))) * t_1)) / t_3;
	} else {
		tmp = t_2 / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1))
	t_3 = fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0))
	tmp = 0.0
	if (x <= -3.8)
		tmp = Float64(t_2 / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	elseif (x <= 0.55)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_0 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_0 / 16.0))) * t_1)) / t_3);
	else
		tmp = Float64(t_2 / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.8], N[(t$95$2 / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.55], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$0 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(t$95$2 / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7999999999999998

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if -3.7999999999999998 < x < 0.55000000000000004

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-pow.f64 51.2

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 51.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{x}^{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-pow.f64 51.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   10. Applied rewrites 51.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if 0.55000000000000004 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{\color{blue}{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-sqrt.f64 64.5

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0}{\mathsf{fma}\left(3 \cdot \cos y, 0.5 \cdot t\_1, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ t_3 := y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\\ \mathbf{if}\;y \leq -0.08:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.86:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{t\_3}{16}\right)\right) \cdot \left(t\_3 - \frac{\sin x}{16}\right)\right) \cdot t\_0}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{t\_1}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (+
           2.0
           (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0))
          (fma
           (* 3.0 (cos y))
           (* 0.5 t_1)
           (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0))))
        (t_3 (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0))))))
   (if (<= y -0.08)
     t_2
     (if (<= y 1.86)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ t_3 16.0))) (- t_3 (/ (sin x) 16.0)))
          t_0))
        (*
         3.0
         (+ (+ 1.0 (* 0.6180339887498949 (cos x))) (* (/ t_1 2.0) (cos y)))))
       t_2))))

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 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / fma((3.0 * cos(y)), (0.5 * t_1), (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	double t_3 = y * (1.0 + (-0.16666666666666666 * pow(y, 2.0)));
	double tmp;
	if (y <= -0.08) {
		tmp = t_2;
	} else if (y <= 1.86) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (t_3 / 16.0))) * (t_3 - (sin(x) / 16.0))) * t_0)) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + ((t_1 / 2.0) * cos(y))));
	} else {
		tmp = t_2;
	}
	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(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0)) / fma(Float64(3.0 * cos(y)), Float64(0.5 * t_1), Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0)))
	t_3 = Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0))))
	tmp = 0.0
	if (y <= -0.08)
		tmp = t_2;
	elseif (y <= 1.86)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(t_3 / 16.0))) * Float64(t_3 - Float64(sin(x) / 16.0))) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(t_1 / 2.0) * cos(y)))));
	else
		tmp = t_2;
	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[(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[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision] + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.08], t$95$2, If[LessEqual[y, 1.86], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(t$95$3 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0800000000000000017 or 1.8600000000000001 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   6. Applied rewrites 64.5%

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

   if -0.0800000000000000017 < y < 1.8600000000000001

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.9

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

   8. Applied rewrites 50.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}{16}\right)\right) \cdot \left(\color{blue}{y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\\ t_1 := 1 + -0.5 \cdot {y}^{2}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\\ t_3 := \frac{2 + \left(t\_0 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), t\_2\right)}\\ \mathbf{if}\;y \leq -0.08:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;\frac{2 + \left(t\_0 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - t\_1\right)}{\mathsf{fma}\left(3 \cdot t\_1, 0.38196601125010515, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))
        (t_1 (+ 1.0 (* -0.5 (pow y 2.0))))
        (t_2 (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0))
        (t_3
         (/
          (+ 2.0 (* (* t_0 (sin y)) (- (cos x) (cos y))))
          (fma (* 3.0 (cos y)) (* 0.5 (- 3.0 (sqrt 5.0))) t_2))))
   (if (<= y -0.08)
     t_3
     (if (<= y 0.035)
       (/
        (+ 2.0 (* (* t_0 (- (sin y) (/ (sin x) 16.0))) (- (cos x) t_1)))
        (fma (* 3.0 t_1) 0.38196601125010515 t_2))
       t_3))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * (sin(x) - (sin(y) / 16.0));
	double t_1 = 1.0 + (-0.5 * pow(y, 2.0));
	double t_2 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0;
	double t_3 = (2.0 + ((t_0 * sin(y)) * (cos(x) - cos(y)))) / fma((3.0 * cos(y)), (0.5 * (3.0 - sqrt(5.0))), t_2);
	double tmp;
	if (y <= -0.08) {
		tmp = t_3;
	} else if (y <= 0.035) {
		tmp = (2.0 + ((t_0 * (sin(y) - (sin(x) / 16.0))) * (cos(x) - t_1))) / fma((3.0 * t_1), 0.38196601125010515, t_2);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))
	t_1 = Float64(1.0 + Float64(-0.5 * (y ^ 2.0)))
	t_2 = Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0)
	t_3 = Float64(Float64(2.0 + Float64(Float64(t_0 * sin(y)) * Float64(cos(x) - cos(y)))) / fma(Float64(3.0 * cos(y)), Float64(0.5 * Float64(3.0 - sqrt(5.0))), t_2))
	tmp = 0.0
	if (y <= -0.08)
		tmp = t_3;
	elseif (y <= 0.035)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_0 * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - t_1))) / fma(Float64(3.0 * t_1), 0.38196601125010515, t_2));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(t$95$0 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.08], t$95$3, If[LessEqual[y, 0.035], N[(N[(2.0 + N[(N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * t$95$1), $MachinePrecision] * 0.38196601125010515 + t$95$2), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0800000000000000017 or 0.035000000000000003 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   6. Applied rewrites 64.5%

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

   if -0.0800000000000000017 < y < 0.035000000000000003

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\left(1 + \frac{-1}{2} \cdot {y}^{2}\right)}\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + \color{blue}{\frac{-1}{2} \cdot {y}^{2}}\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + \frac{-1}{2} \cdot \color{blue}{{y}^{2}}\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 51.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 51.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)}\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + \frac{-1}{2} \cdot {y}^{2}\right)\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {y}^{2}\right)}, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + \frac{-1}{2} \cdot {y}^{2}\right)\right)}{\mathsf{fma}\left(3 \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {y}^{2}}\right), \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + \frac{-1}{2} \cdot {y}^{2}\right)\right)}{\mathsf{fma}\left(3 \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{y}^{2}}\right), \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 53.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot {y}^{2}\right)\right)}{\mathsf{fma}\left(3 \cdot \left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right), 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   10. Applied rewrites 53.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot {y}^{2}\right)\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)}, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 1 + -0.5 \cdot {y}^{2}\\ t_1 := \sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := \frac{2 + \left(t\_1 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.5 \cdot t\_2, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{if}\;y \leq -0.08:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;\frac{2 + \left(t\_1 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - t\_0\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{t\_2}{2} \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -0.5 (pow y 2.0))))
        (t_1 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (/
          (+ 2.0 (* (* t_1 (sin y)) (- (cos x) (cos y))))
          (fma
           (* 3.0 (cos y))
           (* 0.5 t_2)
           (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))))
   (if (<= y -0.08)
     t_3
     (if (<= y 0.035)
       (/
        (+ 2.0 (* (* t_1 (- (sin y) (/ (sin x) 16.0))) (- (cos x) t_0)))
        (* 3.0 (+ (+ 1.0 (* 0.6180339887498949 (cos x))) (* (/ t_2 2.0) t_0))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-0.5 * pow(y, 2.0));
	double t_1 = sqrt(2.0) * (sin(x) - (sin(y) / 16.0));
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = (2.0 + ((t_1 * sin(y)) * (cos(x) - cos(y)))) / fma((3.0 * cos(y)), (0.5 * t_2), (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	double tmp;
	if (y <= -0.08) {
		tmp = t_3;
	} else if (y <= 0.035) {
		tmp = (2.0 + ((t_1 * (sin(y) - (sin(x) / 16.0))) * (cos(x) - t_0))) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + ((t_2 / 2.0) * t_0)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-0.5 * (y ^ 2.0)))
	t_1 = Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(Float64(2.0 + Float64(Float64(t_1 * sin(y)) * Float64(cos(x) - cos(y)))) / fma(Float64(3.0 * cos(y)), Float64(0.5 * t_2), Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0)))
	tmp = 0.0
	if (y <= -0.08)
		tmp = t_3;
	elseif (y <= 0.035)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_1 * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - t_0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(t_2 / 2.0) * t_0))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(t$95$1 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(0.5 * t$95$2), $MachinePrecision] + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.08], t$95$3, If[LessEqual[y, 0.035], N[(N[(2.0 + N[(N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0800000000000000017 or 0.035000000000000003 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   6. Applied rewrites 64.5%

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

   if -0.0800000000000000017 < y < 0.035000000000000003

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.0

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right)\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 51.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)}\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 53.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot {y}^{2}\right)\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + -0.5 \cdot {y}^{\color{blue}{2}}\right)\right)} \]

   8. Applied rewrites 53.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot {y}^{2}\right)\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\left(1 + -0.5 \cdot {y}^{2}\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 \cdot \cos y\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\\ t_3 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0}{\mathsf{fma}\left(t\_1, 0.5 \cdot \left(3 - \sqrt{5}\right), t\_2\right)}\\ \mathbf{if}\;y \leq -0.08:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 7.6:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot y\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(t\_1, 0.38196601125010515, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* 3.0 (cos y)))
        (t_2 (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0))
        (t_3
         (/
          (+
           2.0
           (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0))
          (fma t_1 (* 0.5 (- 3.0 (sqrt 5.0))) t_2))))
   (if (<= y -0.08)
     t_3
     (if (<= y 7.6)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (* 0.0625 y)))
           (- (sin y) (/ (sin x) 16.0)))
          t_0))
        (fma t_1 0.38196601125010515 t_2))
       t_3))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 * cos(y);
	double t_2 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0;
	double t_3 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / fma(t_1, (0.5 * (3.0 - sqrt(5.0))), t_2);
	double tmp;
	if (y <= -0.08) {
		tmp = t_3;
	} else if (y <= 7.6) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (0.0625 * y))) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma(t_1, 0.38196601125010515, t_2);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 * cos(y))
	t_2 = Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0)
	t_3 = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0)) / fma(t_1, Float64(0.5 * Float64(3.0 - sqrt(5.0))), t_2))
	tmp = 0.0
	if (y <= -0.08)
		tmp = t_3;
	elseif (y <= 7.6)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(0.0625 * y))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(t_1, 0.38196601125010515, t_2));
	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[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$3 = 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[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.08], t$95$3, If[LessEqual[y, 7.6], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * 0.38196601125010515 + t$95$2), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0800000000000000017 or 7.5999999999999996 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   6. Applied rewrites 64.5%

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

   if -0.0800000000000000017 < y < 7.5999999999999996

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \color{blue}{\frac{1}{16} \cdot y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 51.8

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot \color{blue}{y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 51.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \color{blue}{0.0625 \cdot y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\\ \mathbf{if}\;y \leq -0.08:\\ \;\;\;\;\frac{t\_1}{3 \cdot \mathsf{fma}\left(t\_2, 0.5 \cdot \cos y, t\_3\right)}\\ \mathbf{elif}\;y \leq 7.6:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot y\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, t\_3 \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(\cos y \cdot t\_2\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (+
          2.0
          (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
   (if (<= y -0.08)
     (/ t_1 (* 3.0 (fma t_2 (* 0.5 (cos y)) t_3)))
     (if (<= y 7.6)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (* 0.0625 y)))
           (- (sin y) (/ (sin x) 16.0)))
          t_0))
        (fma (* 3.0 (cos y)) 0.38196601125010515 (* t_3 3.0)))
       (/
        t_1
        (*
         3.0
         (+
          1.0
          (fma
           0.5
           (* (cos x) (- (sqrt 5.0) 1.0))
           (* 0.5 (* (cos y) t_2))))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0);
	double tmp;
	if (y <= -0.08) {
		tmp = t_1 / (3.0 * fma(t_2, (0.5 * cos(y)), t_3));
	} else if (y <= 7.6) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (0.0625 * y))) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma((3.0 * cos(y)), 0.38196601125010515, (t_3 * 3.0));
	} else {
		tmp = t_1 / (3.0 * (1.0 + fma(0.5, (cos(x) * (sqrt(5.0) - 1.0)), (0.5 * (cos(y) * t_2)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)
	tmp = 0.0
	if (y <= -0.08)
		tmp = Float64(t_1 / Float64(3.0 * fma(t_2, Float64(0.5 * cos(y)), t_3)));
	elseif (y <= 7.6)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(0.0625 * y))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(t_3 * 3.0)));
	else
		tmp = Float64(t_1 / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(sqrt(5.0) - 1.0)), Float64(0.5 * Float64(cos(y) * t_2))))));
	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[(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[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -0.08], N[(t$95$1 / N[(3.0 * N[(t$95$2 * N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.6], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(t$95$3 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0800000000000000017

   1. Initial program 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 99.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.3

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   6. Applied rewrites 64.5%

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

   if -0.0800000000000000017 < y < 7.5999999999999996

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \color{blue}{\frac{1}{16} \cdot y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 51.8

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot \color{blue}{y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 51.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \color{blue}{0.0625 \cdot y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if 7.5999999999999996 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-sin.f64 60.2

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

   7. Applied rewrites 60.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 64.5

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

   10. Applied rewrites 64.5%

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

Alternative 10: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\\ \mathbf{if}\;y \leq -0.08:\\ \;\;\;\;\frac{t\_1}{3 \cdot \mathsf{fma}\left(t\_2, 0.5 \cdot \cos y, t\_3\right)}\\ \mathbf{elif}\;y \leq 7.6:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot y\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, t\_3 \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{t\_2}{2} \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (+
          2.0
          (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
   (if (<= y -0.08)
     (/ t_1 (* 3.0 (fma t_2 (* 0.5 (cos y)) t_3)))
     (if (<= y 7.6)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (* 0.0625 y)))
           (- (sin y) (/ (sin x) 16.0)))
          t_0))
        (fma (* 3.0 (cos y)) 0.38196601125010515 (* t_3 3.0)))
       (/
        t_1
        (*
         3.0
         (+
          (+ 1.0 (* 0.6180339887498949 (cos x)))
          (* (/ t_2 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0);
	double tmp;
	if (y <= -0.08) {
		tmp = t_1 / (3.0 * fma(t_2, (0.5 * cos(y)), t_3));
	} else if (y <= 7.6) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (0.0625 * y))) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma((3.0 * cos(y)), 0.38196601125010515, (t_3 * 3.0));
	} else {
		tmp = t_1 / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + ((t_2 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)
	tmp = 0.0
	if (y <= -0.08)
		tmp = Float64(t_1 / Float64(3.0 * fma(t_2, Float64(0.5 * cos(y)), t_3)));
	elseif (y <= 7.6)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(0.0625 * y))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(t_3 * 3.0)));
	else
		tmp = Float64(t_1 / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(t_2 / 2.0) * 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[(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[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -0.08], N[(t$95$1 / N[(3.0 * N[(t$95$2 * N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.6], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(t$95$3 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0800000000000000017

   1. Initial program 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 99.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.3

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   6. Applied rewrites 64.5%

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

   if -0.0800000000000000017 < y < 7.5999999999999996

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \color{blue}{\frac{1}{16} \cdot y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 51.8

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot \color{blue}{y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 51.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \color{blue}{0.0625 \cdot y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if 7.5999999999999996 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

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

   5. Applied rewrites 64.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0\\ t_2 := 3 - \sqrt{5}\\ t_3 := 3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{t\_2}{2} \cdot \cos y\right)\\ \mathbf{if}\;y \leq -0.08:\\ \;\;\;\;\frac{t\_1}{3 \cdot \mathsf{fma}\left(t\_2, 0.5 \cdot \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 7.6:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot y\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_3}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (+
          2.0
          (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (*
          3.0
          (+ (+ 1.0 (* 0.6180339887498949 (cos x))) (* (/ t_2 2.0) (cos y))))))
   (if (<= y -0.08)
     (/
      t_1
      (*
       3.0
       (fma t_2 (* 0.5 (cos y)) (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))))
     (if (<= y 7.6)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (* 0.0625 y)))
           (- (sin y) (/ (sin x) 16.0)))
          t_0))
        t_3)
       (/ t_1 t_3)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + ((t_2 / 2.0) * cos(y)));
	double tmp;
	if (y <= -0.08) {
		tmp = t_1 / (3.0 * fma(t_2, (0.5 * cos(y)), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)));
	} else if (y <= 7.6) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (0.0625 * y))) * (sin(y) - (sin(x) / 16.0))) * t_0)) / t_3;
	} else {
		tmp = t_1 / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(t_2 / 2.0) * cos(y))))
	tmp = 0.0
	if (y <= -0.08)
		tmp = Float64(t_1 / Float64(3.0 * fma(t_2, Float64(0.5 * cos(y)), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))));
	elseif (y <= 7.6)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(0.0625 * y))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / t_3);
	else
		tmp = Float64(t_1 / 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[(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[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.08], N[(t$95$1 / N[(3.0 * N[(t$95$2 * N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.6], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(t$95$1 / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0800000000000000017

   1. Initial program 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 99.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.3

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   6. Applied rewrites 64.5%

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

   if -0.0800000000000000017 < y < 7.5999999999999996

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \color{blue}{\frac{1}{16} \cdot y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 51.8

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot \color{blue}{y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 51.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \color{blue}{0.0625 \cdot y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 7.5999999999999996 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

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

   5. Applied rewrites 64.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\\ \mathbf{if}\;y \leq -0.08:\\ \;\;\;\;\frac{t\_1}{3 \cdot \mathsf{fma}\left(t\_2, 0.5 \cdot \cos y, t\_3\right)}\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, t\_3 \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{t\_2}{2} \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (+
          2.0
          (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
   (if (<= y -0.08)
     (/ t_1 (* 3.0 (fma t_2 (* 0.5 (cos y)) t_3)))
     (if (<= y 0.035)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_0))
        (fma (* 3.0 (cos y)) 0.38196601125010515 (* t_3 3.0)))
       (/
        t_1
        (*
         3.0
         (+
          (+ 1.0 (* 0.6180339887498949 (cos x)))
          (* (/ t_2 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0);
	double tmp;
	if (y <= -0.08) {
		tmp = t_1 / (3.0 * fma(t_2, (0.5 * cos(y)), t_3));
	} else if (y <= 0.035) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_0)) / fma((3.0 * cos(y)), 0.38196601125010515, (t_3 * 3.0));
	} else {
		tmp = t_1 / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + ((t_2 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)
	tmp = 0.0
	if (y <= -0.08)
		tmp = Float64(t_1 / Float64(3.0 * fma(t_2, Float64(0.5 * cos(y)), t_3)));
	elseif (y <= 0.035)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_0)) / fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(t_3 * 3.0)));
	else
		tmp = Float64(t_1 / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(t_2 / 2.0) * 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[(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[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -0.08], N[(t$95$1 / N[(3.0 * N[(t$95$2 * N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.035], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(t$95$3 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0800000000000000017

   1. Initial program 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 99.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.3

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   6. Applied rewrites 64.5%

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

   if -0.0800000000000000017 < y < 0.035000000000000003

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\color{blue}{y}}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\color{blue}{y}}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(\color{blue}{y} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.0%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(\color{blue}{y} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if 0.035000000000000003 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

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

   5. Applied rewrites 64.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.08:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (/
          (+
           2.0
           (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0))
          (*
           3.0
           (+
            (+ 1.0 (* 0.6180339887498949 (cos x)))
            (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))))
   (if (<= y -0.08)
     t_1
     (if (<= y 0.035)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_0))
        (fma
         (* 3.0 (cos y))
         0.38196601125010515
         (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	double tmp;
	if (y <= -0.08) {
		tmp = t_1;
	} else if (y <= 0.035) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_0)) / fma((3.0 * cos(y)), 0.38196601125010515, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -0.08)
		tmp = t_1;
	elseif (y <= 0.035)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_0)) / fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0)));
	else
		tmp = 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[(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[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.08], t$95$1, If[LessEqual[y, 0.035], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0800000000000000017 or 0.035000000000000003 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if -0.0800000000000000017 < y < 0.035000000000000003

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\color{blue}{y}}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\color{blue}{y}}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(\color{blue}{y} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.0%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(\color{blue}{y} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 1 + -0.5 \cdot {x}^{2}\\ t_1 := \frac{3 - \sqrt{5}}{2} \cdot \cos y\\ t_2 := \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + t\_1\right)}\\ \mathbf{if}\;x \leq -0.74:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(t\_0 - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot t\_0\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -0.5 (pow x 2.0))))
        (t_1 (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))
        (t_2
         (/
          (+
           2.0
           (*
            (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
            (- (cos x) (cos y))))
          (* 3.0 (+ (+ 1.0 (* 0.6180339887498949 (cos x))) t_1)))))
   (if (<= x -0.74)
     t_2
     (if (<= x 0.55)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (* 0.0625 x)))
          (- t_0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* 0.6180339887498949 t_0)) t_1)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-0.5 * pow(x, 2.0));
	double t_1 = ((3.0 - sqrt(5.0)) / 2.0) * cos(y);
	double t_2 = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + t_1));
	double tmp;
	if (x <= -0.74) {
		tmp = t_2;
	} else if (x <= 0.55) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * (t_0 - cos(y)))) / (3.0 * ((1.0 + (0.6180339887498949 * t_0)) + t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + ((-0.5d0) * (x ** 2.0d0))
    t_1 = ((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y)
    t_2 = (2.0d0 + (((sin(x) * sqrt(2.0d0)) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) - cos(y)))) / (3.0d0 * ((1.0d0 + (0.6180339887498949d0 * cos(x))) + t_1))
    if (x <= (-0.74d0)) then
        tmp = t_2
    else if (x <= 0.55d0) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (0.0625d0 * x))) * (t_0 - cos(y)))) / (3.0d0 * ((1.0d0 + (0.6180339887498949d0 * t_0)) + t_1))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (-0.5 * Math.pow(x, 2.0));
	double t_1 = ((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y);
	double t_2 = (2.0 + (((Math.sin(x) * Math.sqrt(2.0)) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) - Math.cos(y)))) / (3.0 * ((1.0 + (0.6180339887498949 * Math.cos(x))) + t_1));
	double tmp;
	if (x <= -0.74) {
		tmp = t_2;
	} else if (x <= 0.55) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (0.0625 * x))) * (t_0 - Math.cos(y)))) / (3.0 * ((1.0 + (0.6180339887498949 * t_0)) + t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (-0.5 * math.pow(x, 2.0))
	t_1 = ((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y)
	t_2 = (2.0 + (((math.sin(x) * math.sqrt(2.0)) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) - math.cos(y)))) / (3.0 * ((1.0 + (0.6180339887498949 * math.cos(x))) + t_1))
	tmp = 0
	if x <= -0.74:
		tmp = t_2
	elif x <= 0.55:
		tmp = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (0.0625 * x))) * (t_0 - math.cos(y)))) / (3.0 * ((1.0 + (0.6180339887498949 * t_0)) + t_1))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-0.5 * (x ^ 2.0)))
	t_1 = Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y))
	t_2 = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + t_1)))
	tmp = 0.0
	if (x <= -0.74)
		tmp = t_2;
	elseif (x <= 0.55)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * x))) * Float64(t_0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * t_0)) + t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (-0.5 * (x ^ 2.0));
	t_1 = ((3.0 - sqrt(5.0)) / 2.0) * cos(y);
	t_2 = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + t_1));
	tmp = 0.0;
	if (x <= -0.74)
		tmp = t_2;
	elseif (x <= 0.55)
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * (t_0 - cos(y)))) / (3.0 * ((1.0 + (0.6180339887498949 * t_0)) + t_1));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.74], t$95$2, If[LessEqual[x, 0.55], 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[(0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.73999999999999999 or 0.55000000000000004 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if -0.73999999999999999 < x < 0.55000000000000004

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{\frac{1}{16} \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 51.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 51.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{0.0625 \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{\color{blue}{2}}\right) - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 51.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 50.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \left(1 + -0.5 \cdot {x}^{\color{blue}{2}}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Applied rewrites 50.9%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)\\ t_2 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.036:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) \cdot t\_0}{t\_1}\\ \mathbf{elif}\;y \leq 1.72:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t\_1}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (fma
          (* 3.0 (cos y))
          0.38196601125010515
          (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))
        (t_2 (pow (sin y) 2.0)))
   (if (<= y -0.036)
     (/ (+ 2.0 (* (* -0.0625 (* t_2 (sqrt 2.0))) t_0)) t_1)
     (if (<= y 1.72)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_0))
        t_1)
       (/ (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) (- 1.0 (cos y)))))) t_1)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma((3.0 * cos(y)), 0.38196601125010515, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	double t_2 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.036) {
		tmp = (2.0 + ((-0.0625 * (t_2 * sqrt(2.0))) * t_0)) / t_1;
	} else if (y <= 1.72) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_0)) / t_1;
	} else {
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(y)))))) / t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0))
	t_2 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.036)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_2 * sqrt(2.0))) * t_0)) / t_1);
	elseif (y <= 1.72)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_0)) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(1.0 - 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[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.036], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 1.72], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0359999999999999973

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{\color{blue}{2}}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sqrt.f64 62.9

         \[\leadsto \frac{2 + \left(-0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if -0.0359999999999999973 < y < 1.71999999999999997

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\color{blue}{y}}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\color{blue}{y}}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(\color{blue}{y} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.0%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(\color{blue}{y} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if 1.71999999999999997 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      8. lower-cos.f64 62.9

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)\\ t_2 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.036:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) \cdot t\_0}{t\_1}\\ \mathbf{elif}\;y \leq 1.72:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t\_1}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (fma
          (* 3.0 (cos y))
          0.38196601125010515
          (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))
        (t_2 (pow (sin y) 2.0)))
   (if (<= y -0.036)
     (/ (+ 2.0 (* (* -0.0625 (* t_2 (sqrt 2.0))) t_0)) t_1)
     (if (<= y 1.72)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_0))
        (*
         3.0
         (+
          (+ 1.0 (* 0.6180339887498949 (cos x)))
          (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
       (/ (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) (- 1.0 (cos y)))))) t_1)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma((3.0 * cos(y)), 0.38196601125010515, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	double t_2 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.036) {
		tmp = (2.0 + ((-0.0625 * (t_2 * sqrt(2.0))) * t_0)) / t_1;
	} else if (y <= 1.72) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_0)) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	} else {
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(y)))))) / t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0))
	t_2 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.036)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_2 * sqrt(2.0))) * t_0)) / t_1);
	elseif (y <= 1.72)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(1.0 - 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[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.036], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 1.72], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0359999999999999973

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{\color{blue}{2}}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sqrt.f64 62.9

         \[\leadsto \frac{2 + \left(-0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if -0.0359999999999999973 < y < 1.71999999999999997

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\color{blue}{y}}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(\color{blue}{y} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.0%

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

   if 1.71999999999999997 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      8. lower-cos.f64 62.9

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) \cdot t\_0}{t\_1}\\ \mathbf{elif}\;x \leq 2.2:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot t\_0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* 0.6180339887498949 (cos x)))
           (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
        (t_2 (pow (sin x) 2.0)))
   (if (<= x -1.9)
     (/ (+ 2.0 (* (* -0.0625 (* t_2 (sqrt 2.0))) t_0)) t_1)
     (if (<= x 2.2)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- x (* 0.0625 (sin y)))) (- (sin y) (* 0.0625 x)))
          t_0))
        t_1)
       (/
        (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) (- (cos x) 1.0)))))
        (fma
         (* 3.0 (cos y))
         0.38196601125010515
         (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y)));
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -1.9) {
		tmp = (2.0 + ((-0.0625 * (t_2 * sqrt(2.0))) * t_0)) / t_1;
	} else if (x <= 2.2) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (0.0625 * x))) * t_0)) / t_1;
	} else {
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * (cos(x) - 1.0))))) / fma((3.0 * cos(y)), 0.38196601125010515, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y))))
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -1.9)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_2 * sqrt(2.0))) * t_0)) / t_1);
	elseif (x <= 2.2)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(0.0625 * sin(y)))) * Float64(sin(y) - Float64(0.0625 * x))) * t_0)) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -1.9], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 2.2], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8999999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.8

         \[\leadsto \frac{2 + \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 62.8%

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

   if -1.8999999999999999 < x < 2.2000000000000002

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{\frac{1}{16} \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 51.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 51.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{0.0625 \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x - \frac{1}{16} \cdot \sin y\right)}\right) \cdot \left(\sin y - \frac{1}{16} \cdot x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.6

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

   8. Applied rewrites 51.6%

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

   if 2.2000000000000002 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      8. lower-cos.f64 62.8

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 62.8%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)\\ t_1 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right)\\ t_2 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -2.9:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{t\_0}\\ \mathbf{elif}\;y \leq 780000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{\mathsf{fma}\left(-0.5, t\_1, -1\right)}, 0.3333333333333333, \frac{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\mathsf{fma}\left(t\_1, 0.5, 1\right)} \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t\_0}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* 3.0 (cos y))
          0.38196601125010515
          (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))
        (t_1 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))))
        (t_2 (pow (sin y) 2.0)))
   (if (<= y -2.9)
     (/ (+ 2.0 (* (* -0.0625 (* t_2 (sqrt 2.0))) (- (cos x) (cos y)))) t_0)
     (if (<= y 780000.0)
       (fma
        (/ -2.0 (fma -0.5 t_1 -1.0))
        0.3333333333333333
        (*
         (/
          (*
           (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
           (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (fma t_1 0.5 1.0))
         0.3333333333333333))
       (/ (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) (- 1.0 (cos y)))))) t_0)))))

C

double code(double x, double y) {
	double t_0 = fma((3.0 * cos(y)), 0.38196601125010515, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	double t_1 = fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0)));
	double t_2 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -2.9) {
		tmp = (2.0 + ((-0.0625 * (t_2 * sqrt(2.0))) * (cos(x) - cos(y)))) / t_0;
	} else if (y <= 780000.0) {
		tmp = fma((-2.0 / fma(-0.5, t_1, -1.0)), 0.3333333333333333, ((((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) / fma(t_1, 0.5, 1.0)) * 0.3333333333333333));
	} else {
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0))
	t_1 = fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0)))
	t_2 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -2.9)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_2 * sqrt(2.0))) * Float64(cos(x) - cos(y)))) / t_0);
	elseif (y <= 780000.0)
		tmp = fma(Float64(-2.0 / fma(-0.5, t_1, -1.0)), 0.3333333333333333, Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) / fma(t_1, 0.5, 1.0)) * 0.3333333333333333));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -2.9], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 780000.0], N[(N[(-2.0 / N[(-0.5 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(N[(N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.89999999999999991

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{\color{blue}{2}}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-sqrt.f64 62.9

         \[\leadsto \frac{2 + \left(-0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if -2.89999999999999991 < y < 7.8e5

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 60.4%

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

   if 7.8e5 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      8. lower-cos.f64 62.9

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ t_1 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right)\\ \mathbf{if}\;y \leq -2.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 780000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{\mathsf{fma}\left(-0.5, t\_1, -1\right)}, 0.3333333333333333, \frac{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\mathsf{fma}\left(t\_1, 0.5, 1\right)} \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
          (fma
           (* 3.0 (cos y))
           0.38196601125010515
           (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0))))
        (t_1 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0)))))
   (if (<= y -2.9)
     t_0
     (if (<= y 780000.0)
       (fma
        (/ -2.0 (fma -0.5 t_1 -1.0))
        0.3333333333333333
        (*
         (/
          (*
           (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
           (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (fma t_1 0.5 1.0))
         0.3333333333333333))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma((3.0 * cos(y)), 0.38196601125010515, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	double t_1 = fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0)));
	double tmp;
	if (y <= -2.9) {
		tmp = t_0;
	} else if (y <= 780000.0) {
		tmp = fma((-2.0 / fma(-0.5, t_1, -1.0)), 0.3333333333333333, ((((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) / fma(t_1, 0.5, 1.0)) * 0.3333333333333333));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0)))
	t_1 = fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0)))
	tmp = 0.0
	if (y <= -2.9)
		tmp = t_0;
	elseif (y <= 780000.0)
		tmp = fma(Float64(-2.0 / fma(-0.5, t_1, -1.0)), 0.3333333333333333, Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) / fma(t_1, 0.5, 1.0)) * 0.3333333333333333));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.9], t$95$0, If[LessEqual[y, 780000.0], N[(N[(-2.0 / N[(-0.5 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(N[(N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.89999999999999991 or 7.8e5 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      8. lower-cos.f64 62.9

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if -2.89999999999999991 < y < 7.8e5

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 60.4%

      \[\leadsto \mathsf{fma}\left(\frac{-2}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}, \color{blue}{0.3333333333333333}, \frac{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)\\ t_1 := \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.32:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.25:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* 3.0 (cos y))
          0.38196601125010515
          (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))
        (t_1
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
          t_0)))
   (if (<= y -0.32)
     t_1
     (if (<= y 0.25)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma((3.0 * cos(y)), 0.38196601125010515, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	double t_1 = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	double tmp;
	if (y <= -0.32) {
		tmp = t_1;
	} else if (y <= 0.25) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0))
	t_1 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_0)
	tmp = 0.0
	if (y <= -0.32)
		tmp = t_1;
	elseif (y <= 0.25)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.32], t$95$1, If[LessEqual[y, 0.25], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.320000000000000007 or 0.25 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      8. lower-cos.f64 62.9

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if -0.320000000000000007 < y < 0.25

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      8. lower-cos.f64 62.8

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 62.8%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.32:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.25:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
          (*
           3.0
           (+
            (+ 1.0 (* 0.6180339887498949 (cos x)))
            (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))))
   (if (<= y -0.32)
     t_0
     (if (<= y 0.25)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (fma
         (* 3.0 (cos y))
         0.38196601125010515
         (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	double tmp;
	if (y <= -0.32) {
		tmp = t_0;
	} else if (y <= 0.25) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma((3.0 * cos(y)), 0.38196601125010515, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -0.32)
		tmp = t_0;
	elseif (y <= 0.25)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(Float64(3.0 * cos(y)), 0.38196601125010515, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.32], t$95$0, If[LessEqual[y, 0.25], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.38196601125010515 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.320000000000000007 or 0.25 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 62.8

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 62.8%

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

   if -0.320000000000000007 < y < 0.25

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, \frac{6880887943736673}{18014398509481984}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

      8. lower-cos.f64 62.8

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   7. Applied rewrites 62.8%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \cos y, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -2.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 780000:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1.1458980337503155 + 3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
          (*
           3.0
           (+
            (+ 1.0 (* 0.6180339887498949 (cos x)))
            (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))))
   (if (<= y -2.9)
     t_0
     (if (<= y 780000.0)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (+
         1.1458980337503155
         (* 3.0 (+ 1.0 (* (cos x) (- (* 0.5 (sqrt 5.0)) 0.5))))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	double tmp;
	if (y <= -2.9) {
		tmp = t_0;
	} else if (y <= 780000.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.1458980337503155 + (3.0 * (1.0 + (cos(x) * ((0.5 * sqrt(5.0)) - 0.5)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 * ((1.0d0 + (0.6180339887498949d0 * cos(x))) + (((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y))))
    if (y <= (-2.9d0)) then
        tmp = t_0
    else if (y <= 780000.0d0) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (1.1458980337503155d0 + (3.0d0 * (1.0d0 + (cos(x) * ((0.5d0 * sqrt(5.0d0)) - 0.5d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 * ((1.0 + (0.6180339887498949 * Math.cos(x))) + (((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y))));
	double tmp;
	if (y <= -2.9) {
		tmp = t_0;
	} else if (y <= 780000.0) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (1.1458980337503155 + (3.0 * (1.0 + (Math.cos(x) * ((0.5 * Math.sqrt(5.0)) - 0.5)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 * ((1.0 + (0.6180339887498949 * math.cos(x))) + (((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y))))
	tmp = 0
	if y <= -2.9:
		tmp = t_0
	elif y <= 780000.0:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (1.1458980337503155 + (3.0 * (1.0 + (math.cos(x) * ((0.5 * math.sqrt(5.0)) - 0.5)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -2.9)
		tmp = t_0;
	elseif (y <= 780000.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(1.1458980337503155 + Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(Float64(0.5 * sqrt(5.0)) - 0.5))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	tmp = 0.0;
	if (y <= -2.9)
		tmp = t_0;
	elseif (y <= 780000.0)
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.1458980337503155 + (3.0 * (1.0 + (cos(x) * ((0.5 * sqrt(5.0)) - 0.5)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.9], t$95$0, If[LessEqual[y, 780000.0], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.1458980337503155 + N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.89999999999999991 or 7.8e5 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 62.8

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 62.8%

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

   if -2.89999999999999991 < y < 7.8e5

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{20642663831210019}{18014398509481984} + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\frac{20642663831210019}{18014398509481984} + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1.1458980337503155 + 3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{5}\\ t_1 := \cos x - 1\\ \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;0.3333333333333333 \cdot \left(\mathsf{fma}\left(0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot \frac{1}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-18}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{1.1458980337503155 + 3 \cdot \left(1 + \cos x \cdot \left(t\_0 - 0.5\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt 5.0))) (t_1 (- (cos x) 1.0)))
   (if (<= x -0.0072)
     (*
      0.3333333333333333
      (*
       (fma (* 0.0625 (* t_1 (sqrt 2.0))) (- 0.5 (* 0.5 (cos (* 2.0 x)))) -2.0)
       (/
        1.0
        (fma -0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) -1.0))))
     (if (<= x 8e-18)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.1458980337503155 (cos y) (* 3.0 (+ 0.5 t_0))))
       (/
        (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_1))))
        (+ 1.1458980337503155 (* 3.0 (+ 1.0 (* (cos x) (- t_0 0.5))))))))))

C

double code(double x, double y) {
	double t_0 = 0.5 * sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double tmp;
	if (x <= -0.0072) {
		tmp = 0.3333333333333333 * (fma((0.0625 * (t_1 * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), -2.0) * (1.0 / fma(-0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), -1.0)));
	} else if (x <= 8e-18) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), (3.0 * (0.5 + t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_1)))) / (1.1458980337503155 + (3.0 * (1.0 + (cos(x) * (t_0 - 0.5)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.5 * sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	tmp = 0.0
	if (x <= -0.0072)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(0.0625 * Float64(t_1 * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), -2.0) * Float64(1.0 / fma(-0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), -1.0))));
	elseif (x <= 8e-18)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), Float64(3.0 * Float64(0.5 + t_0))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_1)))) / Float64(1.1458980337503155 + Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.0072], N[(0.3333333333333333 * N[(N[(N[(0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * N[(1.0 / 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]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e-18], 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[(1.1458980337503155 * N[Cos[y], $MachinePrecision] + N[(3.0 * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.1458980337503155 + N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0071999999999999998

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 60.4%

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

   if -0.0071999999999999998 < x < 8.0000000000000006e-18

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   7. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}} \]

   if 8.0000000000000006e-18 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{20642663831210019}{18014398509481984} + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\frac{20642663831210019}{18014398509481984} + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1.1458980337503155 + 3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right)\\ t_1 := \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)\\ \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;0.3333333333333333 \cdot \left(t\_0 \cdot \frac{1}{t\_1}\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-18}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot 0.3333333333333333}{t\_1}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* 0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x))))
          -2.0))
        (t_1
         (fma -0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) -1.0)))
   (if (<= x -0.0072)
     (* 0.3333333333333333 (* t_0 (/ 1.0 t_1)))
     (if (<= x 8e-18)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.1458980337503155 (cos y) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0))))))
       (/ (* t_0 0.3333333333333333) t_1)))))

C

double code(double x, double y) {
	double t_0 = fma((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), -2.0);
	double t_1 = fma(-0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), -1.0);
	double tmp;
	if (x <= -0.0072) {
		tmp = 0.3333333333333333 * (t_0 * (1.0 / t_1));
	} else if (x <= 8e-18) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	} else {
		tmp = (t_0 * 0.3333333333333333) / t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), -2.0)
	t_1 = fma(-0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), -1.0)
	tmp = 0.0
	if (x <= -0.0072)
		tmp = Float64(0.3333333333333333 * Float64(t_0 * Float64(1.0 / t_1)));
	elseif (x <= 8e-18)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	else
		tmp = Float64(Float64(t_0 * 0.3333333333333333) / t_1);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[x, -0.0072], N[(0.3333333333333333 * N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e-18], 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[(1.1458980337503155 * N[Cos[y], $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 0.3333333333333333), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0071999999999999998

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 60.4%

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

   if -0.0071999999999999998 < x < 8.0000000000000006e-18

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   7. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}} \]

   if 8.0000000000000006e-18 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 60.4%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}\\ \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-18}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (*
           (fma
            (* 0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
            (- 0.5 (* 0.5 (cos (* 2.0 x))))
            -2.0)
           0.3333333333333333)
          (fma
           -0.5
           (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0)))
           -1.0))))
   (if (<= x -0.0072)
     t_0
     (if (<= x 8e-18)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.1458980337503155 (cos y) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0))))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (fma((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), -1.0);
	double tmp;
	if (x <= -0.0072) {
		tmp = t_0;
	} else if (x <= 8e-18) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), -1.0))
	tmp = 0.0
	if (x <= -0.0072)
		tmp = t_0;
	elseif (x <= 8e-18)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * 0.3333333333333333), $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]}, If[LessEqual[x, -0.0072], t$95$0, If[LessEqual[x, 8e-18], 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[(1.1458980337503155 * N[Cos[y], $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0071999999999999998 or 8.0000000000000006e-18 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 60.4%

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

   if -0.0071999999999999998 < x < 8.0000000000000006e-18

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \cos y, \color{blue}{\frac{6880887943736673}{18014398509481984}}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   7. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 60.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), -1.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * 0.3333333333333333), $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]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 60.5%

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

6. Applied rewrites 60.4%

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

Alternative 27: 60.4% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    0.0625
    (* (* (- (cos x) 1.0) (sqrt 2.0)) (- 0.5 (* 0.5 (cos (* 2.0 x)))))
    -2.0)
   (fma -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 - (0.5 * cos((2.0 * x))))), -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(0.5 * cos(Float64(2.0 * x))))), -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[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(-0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 60.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(x + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. lower-sin.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(x + \frac{\mathsf{PI}\left(\right)}{2}\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\frac{\mathsf{PI}\left(\right)}{2} + x\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. mult-flip N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + x\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + x\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. lower-PI.f64 44.2

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\pi, 0.5, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

6. Applied rewrites 44.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\pi, 0.5, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(x + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. lower-sin.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(x + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\frac{\mathsf{PI}\left(\right)}{2} + x\right) \cdot \left(\sqrt{\color{blue}{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. mult-flip N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + x\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + x\right) \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, x\right)\right) \cdot \left(\sqrt{\color{blue}{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. lower-PI.f64 41.8

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\pi, 0.5, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\mathsf{fma}\left(\pi, 0.5, x\right)\right) \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

8. Applied rewrites 41.8%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\sin \left(\mathsf{fma}\left(\pi, 0.5, x\right)\right) - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \sin \left(\mathsf{fma}\left(\pi, 0.5, x\right)\right) \cdot \left(\color{blue}{\sqrt{5}} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

10. Applied rewrites 60.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), -2\right)}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)} \cdot 0.3333333333333333} \]
11. Add Preprocessing

Alternative 28: 43.8% accurate, 5.1× speedup

Math

\[0.3333333333333333 \cdot \frac{2}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   2.0
   (+
    1.0
    (fma 0.5 (* (cos x) (- (sqrt 5.0) 1.0)) (* 0.5 (- 3.0 (sqrt 5.0))))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * (2.0 / (1.0 + fma(0.5, (cos(x) * (sqrt(5.0) - 1.0)), (0.5 * (3.0 - sqrt(5.0))))));
}

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(2.0 / Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(sqrt(5.0) - 1.0)), Float64(0.5 * Float64(3.0 - sqrt(5.0)))))))
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(2.0 / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 60.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \cdot \frac{2}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 43.8%

   \[\leadsto 0.3333333333333333 \cdot \frac{2}{\color{blue}{1} + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
8. Add Preprocessing

Alternative 29: 41.3% accurate, 316.7× speedup

Math

\[0.3333333333333333 \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 60.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{2}{3}}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \color{blue}{\sqrt{5}}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. lower-sqrt.f64 41.3

      \[\leadsto \frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, 3 - \sqrt{5}, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)} \]

7. Applied rewrites 41.3%

   \[\leadsto \frac{0.6666666666666666}{\color{blue}{1 + \mathsf{fma}\left(0.5, 3 - \sqrt{5}, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]

8. Evaluated real constant 41.3%

   \[\leadsto \frac{6004799503160661}{18014398509481984} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025178 
(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))))))