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

Percentage Accurate: 99.3% → 99.4%

Time: 12.6s

Alternatives: 30

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

6. Applied rewrites 99.3%

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

8. Applied rewrites 99.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-+.f64 99.4

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

10. Applied rewrites 99.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

6. Applied rewrites 99.3%

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

8. Applied rewrites 99.3%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.3

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

10. Applied rewrites 99.3%

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

Alternative 3: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot t\_1\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, t\_3, \cos y \cdot t\_0\right)}\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, -0.0625, \sin x\right) \cdot \sqrt{2}, t\_1 \cdot \mathsf{fma}\left(\sin x, -0.0625, t\_2\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_3, \cos x, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot t\_3, \cos x, 1\right), 3, \left(\cos y \cdot \left(0.5 \cdot t\_0\right)\right) \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0)))))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= y -0.15)
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (+ (sin x) (* -0.0625 (sin y))) (* (sin y) t_1))))
      (+ 3.0 (* 1.5 (fma (cos x) t_3 (* (cos y) t_0)))))
     (if (<= y 0.00068)
       (/
        (fma
         (* (fma t_2 -0.0625 (sin x)) (sqrt 2.0))
         (* t_1 (fma (sin x) -0.0625 t_2))
         2.0)
        (fma 1.5 (fma t_3 (cos x) (* t_0 (cos y))) 3.0))
       (/
        (fma t_1 (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0))) 2.0)
        (fma
         (fma (* 0.5 t_3) (cos x) 1.0)
         3.0
         (* (* (cos y) (* 0.5 t_0)) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = y * (1.0 + (-0.16666666666666666 * pow(y, 2.0)));
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.15) {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * t_1)))) / (3.0 + (1.5 * fma(cos(x), t_3, (cos(y) * t_0))));
	} else if (y <= 0.00068) {
		tmp = fma((fma(t_2, -0.0625, sin(x)) * sqrt(2.0)), (t_1 * fma(sin(x), -0.0625, t_2)), 2.0) / fma(1.5, fma(t_3, cos(x), (t_0 * cos(y))), 3.0);
	} else {
		tmp = fma(t_1, (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / fma(fma((0.5 * t_3), cos(x), 1.0), 3.0, ((cos(y) * (0.5 * t_0)) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0))))
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.15)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) * t_1)))) / Float64(3.0 + Float64(1.5 * fma(cos(x), t_3, Float64(cos(y) * t_0)))));
	elseif (y <= 0.00068)
		tmp = Float64(fma(Float64(fma(t_2, -0.0625, sin(x)) * sqrt(2.0)), Float64(t_1 * fma(sin(x), -0.0625, t_2)), 2.0) / fma(1.5, fma(t_3, cos(x), Float64(t_0 * cos(y))), 3.0));
	else
		tmp = Float64(fma(t_1, Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / fma(fma(Float64(0.5 * t_3), cos(x), 1.0), 3.0, Float64(Float64(cos(y) * Float64(0.5 * t_0)) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.15], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$3 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00068], N[(N[(N[(N[(t$95$2 * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + t$95$2), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * t$95$3), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.149999999999999994

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   9. Applied rewrites 63.8%

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

   if -0.149999999999999994 < y < 6.8e-4

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.6

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

   11. Applied rewrites 50.6%

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

   12. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.1

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

   14. Applied rewrites 50.1%

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

   if 6.8e-4 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 63.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-fma.f64 N/A

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

      9. div-add N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l/ N/A

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

   8. Applied rewrites 63.8%

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

Alternative 4: 81.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))
	t_3 = Float64(1.0 + Float64(-0.5 * (y ^ 2.0)))
	t_4 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.011)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) * t_1)))) / Float64(3.0 + Float64(1.5 * fma(cos(x), t_4, Float64(cos(y) * t_0)))));
	elseif (y <= 0.00068)
		tmp = Float64(fma(t_2, Float64(Float64(cos(x) - t_3) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(t_4, cos(x), Float64(t_0 * t_3)), 3.0));
	else
		tmp = Float64(fma(t_1, Float64(sin(y) * t_2), 2.0) / fma(fma(Float64(0.5 * t_4), cos(x), 1.0), 3.0, Float64(Float64(cos(y) * Float64(0.5 * t_0)) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(-0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.011], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$4 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00068], N[(N[(t$95$2 * N[(N[(N[Cos[x], $MachinePrecision] - t$95$3), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$4 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * t$95$4), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.010999999999999999

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   9. Applied rewrites 63.8%

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

   if -0.010999999999999999 < y < 6.8e-4

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 50.2

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

   11. Applied rewrites 50.2%

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

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 52.6

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

   14. Applied rewrites 52.6%

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

   if 6.8e-4 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 63.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-fma.f64 N/A

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

      9. div-add N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l/ N/A

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

   8. Applied rewrites 63.8%

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

Alternative 5: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.05:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot t\_1\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot t\_0\right)}\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sin x + -0.0625 \cdot y\right) \cdot \sqrt{2}, t\_1 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot t\_2, \cos x, 1\right), 3, \left(\cos y \cdot \left(0.5 \cdot t\_0\right)\right) \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= y -0.05)
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (+ (sin x) (* -0.0625 (sin y))) (* (sin y) t_1))))
      (+ 3.0 (* 1.5 (fma (cos x) t_2 (* (cos y) t_0)))))
     (if (<= y 0.00068)
       (/
        (fma
         (* (+ (sin x) (* -0.0625 y)) (sqrt 2.0))
         (* t_1 (fma (sin x) -0.0625 (sin y)))
         2.0)
        (fma 1.5 (fma t_2 (cos x) (* t_0 (cos y))) 3.0))
       (/
        (fma t_1 (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0))) 2.0)
        (fma
         (fma (* 0.5 t_2) (cos x) 1.0)
         3.0
         (* (* (cos y) (* 0.5 t_0)) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.05) {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * t_1)))) / (3.0 + (1.5 * fma(cos(x), t_2, (cos(y) * t_0))));
	} else if (y <= 0.00068) {
		tmp = fma(((sin(x) + (-0.0625 * y)) * sqrt(2.0)), (t_1 * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(t_2, cos(x), (t_0 * cos(y))), 3.0);
	} else {
		tmp = fma(t_1, (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / fma(fma((0.5 * t_2), cos(x), 1.0), 3.0, ((cos(y) * (0.5 * t_0)) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.05)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) * t_1)))) / Float64(3.0 + Float64(1.5 * fma(cos(x), t_2, Float64(cos(y) * t_0)))));
	elseif (y <= 0.00068)
		tmp = Float64(fma(Float64(Float64(sin(x) + Float64(-0.0625 * y)) * sqrt(2.0)), Float64(t_1 * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(t_2, cos(x), Float64(t_0 * cos(y))), 3.0));
	else
		tmp = Float64(fma(t_1, Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / fma(fma(Float64(0.5 * t_2), cos(x), 1.0), 3.0, Float64(Float64(cos(y) * Float64(0.5 * t_0)) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.05], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00068], N[(N[(N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.050000000000000003

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   9. Applied rewrites 63.8%

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

   if -0.050000000000000003 < y < 6.8e-4

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 51.1

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

   11. Applied rewrites 51.1%

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

   if 6.8e-4 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 63.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-fma.f64 N/A

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

      9. div-add N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l/ N/A

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

   8. Applied rewrites 63.8%

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

Alternative 6: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ t_3 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 3\right)\\ \mathbf{if}\;y \leq -0.05:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot t\_1\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot t\_0\right)}\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sin x + -0.0625 \cdot y\right) \cdot \sqrt{2}, t\_1 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, t\_1 \cdot \sin y, 2\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (fma 1.5 (fma t_2 (cos x) (* t_0 (cos y))) 3.0)))
   (if (<= y -0.05)
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (+ (sin x) (* -0.0625 (sin y))) (* (sin y) t_1))))
      (+ 3.0 (* 1.5 (fma (cos x) t_2 (* (cos y) t_0)))))
     (if (<= y 0.00068)
       (/
        (fma
         (* (+ (sin x) (* -0.0625 y)) (sqrt 2.0))
         (* t_1 (fma (sin x) -0.0625 (sin y)))
         2.0)
        t_3)
       (/
        (fma (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)) (* t_1 (sin y)) 2.0)
        t_3)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma(1.5, fma(t_2, cos(x), (t_0 * cos(y))), 3.0);
	double tmp;
	if (y <= -0.05) {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * t_1)))) / (3.0 + (1.5 * fma(cos(x), t_2, (cos(y) * t_0))));
	} else if (y <= 0.00068) {
		tmp = fma(((sin(x) + (-0.0625 * y)) * sqrt(2.0)), (t_1 * fma(sin(x), -0.0625, sin(y))), 2.0) / t_3;
	} else {
		tmp = fma((fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), (t_1 * sin(y)), 2.0) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = fma(1.5, fma(t_2, cos(x), Float64(t_0 * cos(y))), 3.0)
	tmp = 0.0
	if (y <= -0.05)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) * t_1)))) / Float64(3.0 + Float64(1.5 * fma(cos(x), t_2, Float64(cos(y) * t_0)))));
	elseif (y <= 0.00068)
		tmp = Float64(fma(Float64(Float64(sin(x) + Float64(-0.0625 * y)) * sqrt(2.0)), Float64(t_1 * fma(sin(x), -0.0625, sin(y))), 2.0) / t_3);
	else
		tmp = Float64(fma(Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), Float64(t_1 * sin(y)), 2.0) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[y, -0.05], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00068], N[(N[(N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.050000000000000003

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   9. Applied rewrites 63.8%

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

   if -0.050000000000000003 < y < 6.8e-4

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 51.1

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

   11. Applied rewrites 51.1%

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

   if 6.8e-4 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-sin.f64 63.8

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

   11. Applied rewrites 63.8%

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

Alternative 7: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)\\ \mathbf{if}\;x \leq -0.0068:\\ \;\;\;\;-\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\sin x \cdot \sqrt{2}\right), -2\right)}{3 + 3 \cdot \frac{t\_1}{2}}\\ \mathbf{elif}\;x \leq 0.0215:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, x\right) \cdot \sqrt{2}, t\_0 \cdot \mathsf{fma}\left(x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, t\_1, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sqrt{2} \cdot \sin x\right), t\_0, 2\right) \cdot 0.3333333333333333}{1 + t\_1 \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))))
   (if (<= x -0.0068)
     (-
      (/
       (fma
        (- (cos y) (cos x))
        (* (fma (sin x) -0.0625 (sin y)) (* (sin x) (sqrt 2.0)))
        -2.0)
       (+ 3.0 (* 3.0 (/ t_1 2.0)))))
     (if (<= x 0.0215)
       (/
        (fma
         (* (fma (sin y) -0.0625 x) (sqrt 2.0))
         (* t_0 (fma x -0.0625 (sin y)))
         2.0)
        (fma 1.5 t_1 3.0))
       (/
        (*
         (fma
          (* (- (sin y) (* (sin x) 0.0625)) (* (sqrt 2.0) (sin x)))
          t_0
          2.0)
         0.3333333333333333)
        (+ 1.0 (* t_1 0.5)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y)));
	double tmp;
	if (x <= -0.0068) {
		tmp = -(fma((cos(y) - cos(x)), (fma(sin(x), -0.0625, sin(y)) * (sin(x) * sqrt(2.0))), -2.0) / (3.0 + (3.0 * (t_1 / 2.0))));
	} else if (x <= 0.0215) {
		tmp = fma((fma(sin(y), -0.0625, x) * sqrt(2.0)), (t_0 * fma(x, -0.0625, sin(y))), 2.0) / fma(1.5, t_1, 3.0);
	} else {
		tmp = (fma(((sin(y) - (sin(x) * 0.0625)) * (sqrt(2.0) * sin(x))), t_0, 2.0) * 0.3333333333333333) / (1.0 + (t_1 * 0.5));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y)))
	tmp = 0.0
	if (x <= -0.0068)
		tmp = Float64(-Float64(fma(Float64(cos(y) - cos(x)), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(sin(x) * sqrt(2.0))), -2.0) / Float64(3.0 + Float64(3.0 * Float64(t_1 / 2.0)))));
	elseif (x <= 0.0215)
		tmp = Float64(fma(Float64(fma(sin(y), -0.0625, x) * sqrt(2.0)), Float64(t_0 * fma(x, -0.0625, sin(y))), 2.0) / fma(1.5, t_1, 3.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sqrt(2.0) * sin(x))), t_0, 2.0) * 0.3333333333333333) / Float64(1.0 + Float64(t_1 * 0.5)));
	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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0068], (-N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[x, 0.0215], N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(x * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * t$95$1 + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(1.0 + N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00679999999999999962

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.0

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

   6. Applied rewrites 64.0%

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

   if -0.00679999999999999962 < x < 0.021499999999999998

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 50.7%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 50.8%

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

   if 0.021499999999999998 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.0

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

   4. Applied rewrites 64.0%

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

   6. Applied rewrites 64.1%

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

Alternative 8: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sqrt{2} \cdot \sin x\right), t\_0, 2\right)\\ t_2 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)\\ t_3 := 1 + t\_2 \cdot 0.5\\ \mathbf{if}\;x \leq -0.0068:\\ \;\;\;\;\frac{t\_1}{t\_3 \cdot 3}\\ \mathbf{elif}\;x \leq 0.0215:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, x\right) \cdot \sqrt{2}, t\_0 \cdot \mathsf{fma}\left(x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, t\_2, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot 0.3333333333333333}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (fma
          (* (- (sin y) (* (sin x) 0.0625)) (* (sqrt 2.0) (sin x)))
          t_0
          2.0))
        (t_2 (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))))
        (t_3 (+ 1.0 (* t_2 0.5))))
   (if (<= x -0.0068)
     (/ t_1 (* t_3 3.0))
     (if (<= x 0.0215)
       (/
        (fma
         (* (fma (sin y) -0.0625 x) (sqrt 2.0))
         (* t_0 (fma x -0.0625 (sin y)))
         2.0)
        (fma 1.5 t_2 3.0))
       (/ (* t_1 0.3333333333333333) t_3)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma(((sin(y) - (sin(x) * 0.0625)) * (sqrt(2.0) * sin(x))), t_0, 2.0);
	double t_2 = fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y)));
	double t_3 = 1.0 + (t_2 * 0.5);
	double tmp;
	if (x <= -0.0068) {
		tmp = t_1 / (t_3 * 3.0);
	} else if (x <= 0.0215) {
		tmp = fma((fma(sin(y), -0.0625, x) * sqrt(2.0)), (t_0 * fma(x, -0.0625, sin(y))), 2.0) / fma(1.5, t_2, 3.0);
	} else {
		tmp = (t_1 * 0.3333333333333333) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sqrt(2.0) * sin(x))), t_0, 2.0)
	t_2 = fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y)))
	t_3 = Float64(1.0 + Float64(t_2 * 0.5))
	tmp = 0.0
	if (x <= -0.0068)
		tmp = Float64(t_1 / Float64(t_3 * 3.0));
	elseif (x <= 0.0215)
		tmp = Float64(fma(Float64(fma(sin(y), -0.0625, x) * sqrt(2.0)), Float64(t_0 * fma(x, -0.0625, sin(y))), 2.0) / fma(1.5, t_2, 3.0));
	else
		tmp = Float64(Float64(t_1 * 0.3333333333333333) / 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[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0068], N[(t$95$1 / N[(t$95$3 * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0215], N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(x * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * t$95$2 + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * 0.3333333333333333), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00679999999999999962

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.0

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

   4. Applied rewrites 64.0%

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

   6. Applied rewrites 64.0%

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

   if -0.00679999999999999962 < x < 0.021499999999999998

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 50.7%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 50.8%

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

   if 0.021499999999999998 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.0

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

   4. Applied rewrites 64.0%

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

   6. Applied rewrites 64.1%

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

Alternative 9: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)\\ t_2 := \frac{\mathsf{fma}\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sqrt{2} \cdot \sin x\right), t\_0, 2\right) \cdot 0.3333333333333333}{1 + t\_1 \cdot 0.5}\\ \mathbf{if}\;x \leq -0.0068:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.0215:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, x\right) \cdot \sqrt{2}, t\_0 \cdot \mathsf{fma}\left(x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, t\_1, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))))
        (t_2
         (/
          (*
           (fma
            (* (- (sin y) (* (sin x) 0.0625)) (* (sqrt 2.0) (sin x)))
            t_0
            2.0)
           0.3333333333333333)
          (+ 1.0 (* t_1 0.5)))))
   (if (<= x -0.0068)
     t_2
     (if (<= x 0.0215)
       (/
        (fma
         (* (fma (sin y) -0.0625 x) (sqrt 2.0))
         (* t_0 (fma x -0.0625 (sin y)))
         2.0)
        (fma 1.5 t_1 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y)));
	double t_2 = (fma(((sin(y) - (sin(x) * 0.0625)) * (sqrt(2.0) * sin(x))), t_0, 2.0) * 0.3333333333333333) / (1.0 + (t_1 * 0.5));
	double tmp;
	if (x <= -0.0068) {
		tmp = t_2;
	} else if (x <= 0.0215) {
		tmp = fma((fma(sin(y), -0.0625, x) * sqrt(2.0)), (t_0 * fma(x, -0.0625, sin(y))), 2.0) / fma(1.5, t_1, 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y)))
	t_2 = Float64(Float64(fma(Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sqrt(2.0) * sin(x))), t_0, 2.0) * 0.3333333333333333) / Float64(1.0 + Float64(t_1 * 0.5)))
	tmp = 0.0
	if (x <= -0.0068)
		tmp = t_2;
	elseif (x <= 0.0215)
		tmp = Float64(fma(Float64(fma(sin(y), -0.0625, x) * sqrt(2.0)), Float64(t_0 * fma(x, -0.0625, sin(y))), 2.0) / fma(1.5, t_1, 3.0));
	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[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(1.0 + N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0068], t$95$2, If[LessEqual[x, 0.0215], N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(x * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * t$95$1 + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00679999999999999962 or 0.021499999999999998 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.0

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

   4. Applied rewrites 64.0%

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

   6. Applied rewrites 64.1%

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

   if -0.00679999999999999962 < x < 0.021499999999999998

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 50.7%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 50.8%

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

Alternative 10: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ t_3 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 3\right)\\ \mathbf{if}\;y \leq -0.014:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot t\_1\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot t\_0\right)}\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \sqrt{2}, t\_1 \cdot \mathsf{fma}\left(\sin x, -0.0625, y\right), 2\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, t\_1 \cdot \sin y, 2\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (fma 1.5 (fma t_2 (cos x) (* t_0 (cos y))) 3.0)))
   (if (<= y -0.014)
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (+ (sin x) (* -0.0625 (sin y))) (* (sin y) t_1))))
      (+ 3.0 (* 1.5 (fma (cos x) t_2 (* (cos y) t_0)))))
     (if (<= y 0.00068)
       (/
        (fma
         (* (fma y -0.0625 (sin x)) (sqrt 2.0))
         (* t_1 (fma (sin x) -0.0625 y))
         2.0)
        t_3)
       (/
        (fma (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)) (* t_1 (sin y)) 2.0)
        t_3)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma(1.5, fma(t_2, cos(x), (t_0 * cos(y))), 3.0);
	double tmp;
	if (y <= -0.014) {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * t_1)))) / (3.0 + (1.5 * fma(cos(x), t_2, (cos(y) * t_0))));
	} else if (y <= 0.00068) {
		tmp = fma((fma(y, -0.0625, sin(x)) * sqrt(2.0)), (t_1 * fma(sin(x), -0.0625, y)), 2.0) / t_3;
	} else {
		tmp = fma((fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), (t_1 * sin(y)), 2.0) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = fma(1.5, fma(t_2, cos(x), Float64(t_0 * cos(y))), 3.0)
	tmp = 0.0
	if (y <= -0.014)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) * t_1)))) / Float64(3.0 + Float64(1.5 * fma(cos(x), t_2, Float64(cos(y) * t_0)))));
	elseif (y <= 0.00068)
		tmp = Float64(fma(Float64(fma(y, -0.0625, sin(x)) * sqrt(2.0)), Float64(t_1 * fma(sin(x), -0.0625, y)), 2.0) / t_3);
	else
		tmp = Float64(fma(Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), Float64(t_1 * sin(y)), 2.0) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[y, -0.014], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00068], N[(N[(N[(N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + y), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0140000000000000003

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   9. Applied rewrites 63.8%

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

   if -0.0140000000000000003 < y < 6.8e-4

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 51.1%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 50.2%

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

   if 6.8e-4 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-sin.f64 63.8

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

   11. Applied rewrites 63.8%

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

Alternative 11: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, t\_0 \cdot \sin y, 2\right)}{t\_1}\\ \mathbf{if}\;y \leq -0.014:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \sqrt{2}, t\_0 \cdot \mathsf{fma}\left(\sin x, -0.0625, y\right), 2\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (fma
          1.5
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          3.0))
        (t_2
         (/
          (fma
           (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0))
           (* t_0 (sin y))
           2.0)
          t_1)))
   (if (<= y -0.014)
     t_2
     (if (<= y 0.00068)
       (/
        (fma
         (* (fma y -0.0625 (sin x)) (sqrt 2.0))
         (* t_0 (fma (sin x) -0.0625 y))
         2.0)
        t_1)
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	double t_2 = fma((fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), (t_0 * sin(y)), 2.0) / t_1;
	double tmp;
	if (y <= -0.014) {
		tmp = t_2;
	} else if (y <= 0.00068) {
		tmp = fma((fma(y, -0.0625, sin(x)) * sqrt(2.0)), (t_0 * fma(sin(x), -0.0625, y)), 2.0) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0)
	t_2 = Float64(fma(Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), Float64(t_0 * sin(y)), 2.0) / t_1)
	tmp = 0.0
	if (y <= -0.014)
		tmp = t_2;
	elseif (y <= 0.00068)
		tmp = Float64(fma(Float64(fma(y, -0.0625, sin(x)) * sqrt(2.0)), Float64(t_0 * fma(sin(x), -0.0625, y)), 2.0) / t_1);
	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[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[y, -0.014], t$95$2, If[LessEqual[y, 0.00068], N[(N[(N[(N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + y), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0140000000000000003 or 6.8e-4 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-sin.f64 63.8

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

   11. Applied rewrites 63.8%

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

   if -0.0140000000000000003 < y < 6.8e-4

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 51.1%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 50.2%

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

Alternative 12: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sin y \cdot \left(1 - \cos y\right)\\ t_2 := \sqrt{5} - 1\\ t_3 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 3\right)\\ \mathbf{if}\;y \leq -0.014:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot t\_1\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot t\_0\right)}\\ \mathbf{elif}\;y \leq 0.00068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \sqrt{2}, \left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, y\right), 2\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, t\_1, 2\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (sin y) (- 1.0 (cos y))))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (fma 1.5 (fma t_2 (cos x) (* t_0 (cos y))) 3.0)))
   (if (<= y -0.014)
     (/
      (+ 2.0 (* (sqrt 2.0) (* (+ (sin x) (* -0.0625 (sin y))) t_1)))
      (+ 3.0 (* 1.5 (fma (cos x) t_2 (* (cos y) t_0)))))
     (if (<= y 0.00068)
       (/
        (fma
         (* (fma y -0.0625 (sin x)) (sqrt 2.0))
         (* (- (cos x) (cos y)) (fma (sin x) -0.0625 y))
         2.0)
        t_3)
       (/ (fma (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)) t_1 2.0) t_3)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sin(y) * (1.0 - cos(y));
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma(1.5, fma(t_2, cos(x), (t_0 * cos(y))), 3.0);
	double tmp;
	if (y <= -0.014) {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * t_1))) / (3.0 + (1.5 * fma(cos(x), t_2, (cos(y) * t_0))));
	} else if (y <= 0.00068) {
		tmp = fma((fma(y, -0.0625, sin(x)) * sqrt(2.0)), ((cos(x) - cos(y)) * fma(sin(x), -0.0625, y)), 2.0) / t_3;
	} else {
		tmp = fma((fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), t_1, 2.0) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sin(y) * Float64(1.0 - cos(y)))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = fma(1.5, fma(t_2, cos(x), Float64(t_0 * cos(y))), 3.0)
	tmp = 0.0
	if (y <= -0.014)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * t_1))) / Float64(3.0 + Float64(1.5 * fma(cos(x), t_2, Float64(cos(y) * t_0)))));
	elseif (y <= 0.00068)
		tmp = Float64(fma(Float64(fma(y, -0.0625, sin(x)) * sqrt(2.0)), Float64(Float64(cos(x) - cos(y)) * fma(sin(x), -0.0625, y)), 2.0) / t_3);
	else
		tmp = Float64(fma(Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), t_1, 2.0) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[y, -0.014], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00068], N[(N[(N[(N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + y), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0140000000000000003

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 62.1

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

   9. Applied rewrites 62.1%

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

   if -0.0140000000000000003 < y < 6.8e-4

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 51.1%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 50.2%

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

   if 6.8e-4 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-cos.f64 62.1

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

   11. Applied rewrites 62.1%

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

Alternative 13: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ t_3 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_4 := \mathsf{fma}\left(t\_1, \cos x, t\_3\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, -0.0625 \cdot t\_0, 2\right)}{3 + 3 \cdot \frac{t\_4}{2}}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, \left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, 1, t\_3\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, -0.0625 \cdot \left(t\_0 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, t\_4, 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_4 (fma t_1 (cos x) t_3)))
   (if (<= x -3.8e-6)
     (/
      (fma (* t_2 (sqrt 2.0)) (* -0.0625 t_0) 2.0)
      (+ 3.0 (* 3.0 (/ t_4 2.0))))
     (if (<= x 2.5e-5)
       (/
        (fma
         (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0))
         (* (- 1.0 (cos y)) (fma (sin x) -0.0625 (sin y)))
         2.0)
        (fma 1.5 (fma t_1 1.0 t_3) 3.0))
       (/
        (fma t_2 (* -0.0625 (* t_0 (sqrt 2.0))) 2.0)
        (* (fma 0.5 t_4 1.0) 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = (3.0 - sqrt(5.0)) * cos(y);
	double t_4 = fma(t_1, cos(x), t_3);
	double tmp;
	if (x <= -3.8e-6) {
		tmp = fma((t_2 * sqrt(2.0)), (-0.0625 * t_0), 2.0) / (3.0 + (3.0 * (t_4 / 2.0)));
	} else if (x <= 2.5e-5) {
		tmp = fma((fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), ((1.0 - cos(y)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(t_1, 1.0, t_3), 3.0);
	} else {
		tmp = fma(t_2, (-0.0625 * (t_0 * sqrt(2.0))), 2.0) / (fma(0.5, t_4, 1.0) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_4 = fma(t_1, cos(x), t_3)
	tmp = 0.0
	if (x <= -3.8e-6)
		tmp = Float64(fma(Float64(t_2 * sqrt(2.0)), Float64(-0.0625 * t_0), 2.0) / Float64(3.0 + Float64(3.0 * Float64(t_4 / 2.0))));
	elseif (x <= 2.5e-5)
		tmp = Float64(fma(Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), Float64(Float64(1.0 - cos(y)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(t_1, 1.0, t_3), 3.0));
	else
		tmp = Float64(fma(t_2, Float64(-0.0625 * Float64(t_0 * sqrt(2.0))), 2.0) / Float64(fma(0.5, t_4, 1.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[x, -3.8e-6], N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$4 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-5], N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$1 * 1.0 + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(-0.0625 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$4 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.3

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

   6. Applied rewrites 62.3%

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

   if -3.8e-6 < x < 2.50000000000000012e-5

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 62.2%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 59.5%

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

   if 2.50000000000000012e-5 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3

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

   9. Applied rewrites 62.3%

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

Alternative 14: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sin y \cdot \left(1 - \cos y\right)\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot t\_1\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot t\_0\right)}\\ \mathbf{elif}\;y \leq 0.00044:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\sqrt{5} - -1}, 0.5 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (sin y) (- 1.0 (cos y))))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= y -3.3e-6)
     (/
      (+ 2.0 (* (sqrt 2.0) (* (+ (sin x) (* -0.0625 (sin y))) t_1)))
      (+ 3.0 (* 1.5 (fma (cos x) t_2 (* (cos y) t_0)))))
     (if (<= y 0.00044)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (+
          1.0
          (fma 0.5 (* (cos x) (/ 4.0 (- (sqrt 5.0) -1.0))) (* 0.5 t_0)))))
       (/
        (fma (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)) t_1 2.0)
        (fma 1.5 (fma t_2 (cos x) (* t_0 (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sin(y) * (1.0 - cos(y));
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -3.3e-6) {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * t_1))) / (3.0 + (1.5 * fma(cos(x), t_2, (cos(y) * t_0))));
	} else if (y <= 0.00044) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.0 + fma(0.5, (cos(x) * (4.0 / (sqrt(5.0) - -1.0))), (0.5 * t_0))));
	} else {
		tmp = fma((fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), t_1, 2.0) / fma(1.5, fma(t_2, cos(x), (t_0 * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sin(y) * Float64(1.0 - cos(y)))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -3.3e-6)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * t_1))) / Float64(3.0 + Float64(1.5 * fma(cos(x), t_2, Float64(cos(y) * t_0)))));
	elseif (y <= 0.00044)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(4.0 / Float64(sqrt(5.0) - -1.0))), Float64(0.5 * t_0)))));
	else
		tmp = Float64(fma(Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), t_1, 2.0) / fma(1.5, fma(t_2, cos(x), Float64(t_0 * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -3.3e-6], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00044], N[(0.3333333333333333 * 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[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.30000000000000017e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 62.1

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

   9. Applied rewrites 62.1%

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

   if -3.30000000000000017e-6 < y < 4.40000000000000016e-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. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 60.0

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\sqrt{5} - -1}, 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 60.0%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\color{blue}{\sqrt{5} - -1}}, 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if 4.40000000000000016e-4 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-cos.f64 62.1

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

   11. Applied rewrites 62.1%

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

Alternative 15: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.00102:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, -0.0625 \cdot t\_0, 2\right)}{3 + 3 \cdot \frac{t\_1}{2}}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, \sin y \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, t\_1, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, -0.0625 \cdot \left(t\_0 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, t\_1, 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))))
        (t_2 (- (cos x) (cos y))))
   (if (<= x -0.00102)
     (/
      (fma (* t_2 (sqrt 2.0)) (* -0.0625 t_0) 2.0)
      (+ 3.0 (* 3.0 (/ t_1 2.0))))
     (if (<= x 2.5e-5)
       (/
        (fma
         (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0))
         (* (sin y) (- 1.0 (cos y)))
         2.0)
        (fma 1.5 t_1 3.0))
       (/
        (fma t_2 (* -0.0625 (* t_0 (sqrt 2.0))) 2.0)
        (* (fma 0.5 t_1 1.0) 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y)));
	double t_2 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.00102) {
		tmp = fma((t_2 * sqrt(2.0)), (-0.0625 * t_0), 2.0) / (3.0 + (3.0 * (t_1 / 2.0)));
	} else if (x <= 2.5e-5) {
		tmp = fma((fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), (sin(y) * (1.0 - cos(y))), 2.0) / fma(1.5, t_1, 3.0);
	} else {
		tmp = fma(t_2, (-0.0625 * (t_0 * sqrt(2.0))), 2.0) / (fma(0.5, t_1, 1.0) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y)))
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.00102)
		tmp = Float64(fma(Float64(t_2 * sqrt(2.0)), Float64(-0.0625 * t_0), 2.0) / Float64(3.0 + Float64(3.0 * Float64(t_1 / 2.0))));
	elseif (x <= 2.5e-5)
		tmp = Float64(fma(Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), Float64(sin(y) * Float64(1.0 - cos(y))), 2.0) / fma(1.5, t_1, 3.0));
	else
		tmp = Float64(fma(t_2, Float64(-0.0625 * Float64(t_0 * sqrt(2.0))), 2.0) / Float64(fma(0.5, t_1, 1.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00102], N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-5], N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * t$95$1 + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(-0.0625 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$1 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00102

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.3

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

   6. Applied rewrites 62.3%

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

   if -0.00102 < x < 2.50000000000000012e-5

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-cos.f64 62.1

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

   11. Applied rewrites 62.1%

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

   if 2.50000000000000012e-5 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3

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

   9. Applied rewrites 62.3%

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

Alternative 16: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ t_3 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_4 := \mathsf{fma}\left(t\_1, \cos x, t\_3\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, -0.0625 \cdot t\_0, 2\right)}{3 + 3 \cdot \frac{t\_4}{2}}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, 1, t\_3\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, -0.0625 \cdot \left(t\_0 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, t\_4, 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_4 (fma t_1 (cos x) t_3)))
   (if (<= x -3.8e-6)
     (/
      (fma (* t_2 (sqrt 2.0)) (* -0.0625 t_0) 2.0)
      (+ 3.0 (* 3.0 (/ t_4 2.0))))
     (if (<= x 1.2e-5)
       (/
        (fma
         (- 1.0 (cos y))
         (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
         2.0)
        (* (fma 0.5 (fma t_1 1.0 t_3) 1.0) 3.0))
       (/
        (fma t_2 (* -0.0625 (* t_0 (sqrt 2.0))) 2.0)
        (* (fma 0.5 t_4 1.0) 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = (3.0 - sqrt(5.0)) * cos(y);
	double t_4 = fma(t_1, cos(x), t_3);
	double tmp;
	if (x <= -3.8e-6) {
		tmp = fma((t_2 * sqrt(2.0)), (-0.0625 * t_0), 2.0) / (3.0 + (3.0 * (t_4 / 2.0)));
	} else if (x <= 1.2e-5) {
		tmp = fma((1.0 - cos(y)), (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / (fma(0.5, fma(t_1, 1.0, t_3), 1.0) * 3.0);
	} else {
		tmp = fma(t_2, (-0.0625 * (t_0 * sqrt(2.0))), 2.0) / (fma(0.5, t_4, 1.0) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_4 = fma(t_1, cos(x), t_3)
	tmp = 0.0
	if (x <= -3.8e-6)
		tmp = Float64(fma(Float64(t_2 * sqrt(2.0)), Float64(-0.0625 * t_0), 2.0) / Float64(3.0 + Float64(3.0 * Float64(t_4 / 2.0))));
	elseif (x <= 1.2e-5)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / Float64(fma(0.5, fma(t_1, 1.0, t_3), 1.0) * 3.0));
	else
		tmp = Float64(fma(t_2, Float64(-0.0625 * Float64(t_0 * sqrt(2.0))), 2.0) / Float64(fma(0.5, t_4, 1.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[x, -3.8e-6], N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$4 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-5], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * 1.0 + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(-0.0625 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$4 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.3

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

   6. Applied rewrites 62.3%

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

   if -3.8e-6 < x < 1.2e-5

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 62.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 59.4%

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

   if 1.2e-5 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3

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

   9. Applied rewrites 62.3%

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

Alternative 17: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_2 := \frac{\mathsf{fma}\left(\cos x - \cos y, -0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right) \cdot 3}\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, 1, t\_1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_2
         (/
          (fma
           (- (cos x) (cos y))
           (* -0.0625 (* (pow (sin x) 2.0) (sqrt 2.0)))
           2.0)
          (* (fma 0.5 (fma t_0 (cos x) t_1) 1.0) 3.0))))
   (if (<= x -3.8e-6)
     t_2
     (if (<= x 1.2e-5)
       (/
        (fma
         (- 1.0 (cos y))
         (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
         2.0)
        (* (fma 0.5 (fma t_0 1.0 t_1) 1.0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (3.0 - sqrt(5.0)) * cos(y);
	double t_2 = fma((cos(x) - cos(y)), (-0.0625 * (pow(sin(x), 2.0) * sqrt(2.0))), 2.0) / (fma(0.5, fma(t_0, cos(x), t_1), 1.0) * 3.0);
	double tmp;
	if (x <= -3.8e-6) {
		tmp = t_2;
	} else if (x <= 1.2e-5) {
		tmp = fma((1.0 - cos(y)), (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / (fma(0.5, fma(t_0, 1.0, t_1), 1.0) * 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_2 = Float64(fma(Float64(cos(x) - cos(y)), Float64(-0.0625 * Float64((sin(x) ^ 2.0) * sqrt(2.0))), 2.0) / Float64(fma(0.5, fma(t_0, cos(x), t_1), 1.0) * 3.0))
	tmp = 0.0
	if (x <= -3.8e-6)
		tmp = t_2;
	elseif (x <= 1.2e-5)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / Float64(fma(0.5, fma(t_0, 1.0, t_1), 1.0) * 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.8e-6], t$95$2, If[LessEqual[x, 1.2e-5], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e-6 or 1.2e-5 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3

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

   9. Applied rewrites 62.3%

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

   if -3.8e-6 < x < 1.2e-5

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 62.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 59.4%

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

Alternative 18: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := t\_2 \cdot \cos y\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot t\_0\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, \frac{4}{\sqrt{5} - -1}, \cos y \cdot t\_2\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, 1, t\_3\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot t\_0, \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos x, t\_3\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (* t_2 (cos y))))
   (if (<= x -3.8e-6)
     (/
      (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (pow (sin x) 2.0) t_0))))
      (+
       3.0
       (* 1.5 (fma (cos x) (/ 4.0 (- (sqrt 5.0) -1.0)) (* (cos y) t_2)))))
     (if (<= x 1.2e-5)
       (/
        (fma
         (- 1.0 (cos y))
         (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
         2.0)
        (* (fma 0.5 (fma t_1 1.0 t_3) 1.0) 3.0))
       (/
        (fma
         (* (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_0)
         (sqrt 2.0)
         2.0)
        (fma 1.5 (fma t_1 (cos x) t_3) 3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = t_2 * cos(y);
	double tmp;
	if (x <= -3.8e-6) {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(x), 2.0) * t_0)))) / (3.0 + (1.5 * fma(cos(x), (4.0 / (sqrt(5.0) - -1.0)), (cos(y) * t_2))));
	} else if (x <= 1.2e-5) {
		tmp = fma((1.0 - cos(y)), (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / (fma(0.5, fma(t_1, 1.0, t_3), 1.0) * 3.0);
	} else {
		tmp = fma(((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))) * t_0), sqrt(2.0), 2.0) / fma(1.5, fma(t_1, cos(x), t_3), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(t_2 * cos(y))
	tmp = 0.0
	if (x <= -3.8e-6)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(x) ^ 2.0) * t_0)))) / Float64(3.0 + Float64(1.5 * fma(cos(x), Float64(4.0 / Float64(sqrt(5.0) - -1.0)), Float64(cos(y) * t_2)))));
	elseif (x <= 1.2e-5)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / Float64(fma(0.5, fma(t_1, 1.0, t_3), 1.0) * 3.0));
	else
		tmp = Float64(fma(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) * t_0), sqrt(2.0), 2.0) / fma(1.5, fma(t_1, cos(x), t_3), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.8e-6], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-5], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * 1.0 + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.3

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

   9. Applied rewrites 62.3%

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

       1. lift--.f64 N/A

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

       2. flip-- N/A

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

       3. metadata-eval N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. rem-square-sqrt N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. metadata-eval N/A

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

       11. add-flip N/A

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

       12. lower--.f64 N/A

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

       13. metadata-eval 62.3

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

   11. Applied rewrites 62.3%

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

   if -3.8e-6 < x < 1.2e-5

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 62.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 59.4%

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

   if 1.2e-5 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.3

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

   9. Applied rewrites 62.3%

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

   11. Applied rewrites 62.3%

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

Alternative 19: 79.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.0008:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot t\_0\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, \frac{4}{\sqrt{5} - -1}, \cos y \cdot t\_1\right)}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;-\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \mathsf{fma}\left(0.5 \cdot t\_1, \cos y, \cos x \cdot \left(0.5 \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot t\_0, \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_1 \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.0008)
     (/
      (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (pow (sin x) 2.0) t_0))))
      (+
       3.0
       (* 1.5 (fma (cos x) (/ 4.0 (- (sqrt 5.0) -1.0)) (* (cos y) t_1)))))
     (if (<= x 2.5e-5)
       (-
        (/
         (-
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- (cos y) 1.0))))
          2.0)
         (+ 3.0 (* 3.0 (fma (* 0.5 t_1) (cos y) (* (cos x) (* 0.5 t_2)))))))
       (/
        (fma
         (* (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_0)
         (sqrt 2.0)
         2.0)
        (fma 1.5 (fma t_2 (cos x) (* t_1 (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.0008) {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(x), 2.0) * t_0)))) / (3.0 + (1.5 * fma(cos(x), (4.0 / (sqrt(5.0) - -1.0)), (cos(y) * t_1))));
	} else if (x <= 2.5e-5) {
		tmp = -(((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / (3.0 + (3.0 * fma((0.5 * t_1), cos(y), (cos(x) * (0.5 * t_2))))));
	} else {
		tmp = fma(((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))) * t_0), sqrt(2.0), 2.0) / fma(1.5, fma(t_2, cos(x), (t_1 * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.0008)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(x) ^ 2.0) * t_0)))) / Float64(3.0 + Float64(1.5 * fma(cos(x), Float64(4.0 / Float64(sqrt(5.0) - -1.0)), Float64(cos(y) * t_1)))));
	elseif (x <= 2.5e-5)
		tmp = Float64(-Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(3.0 + Float64(3.0 * fma(Float64(0.5 * t_1), cos(y), Float64(cos(x) * Float64(0.5 * t_2)))))));
	else
		tmp = Float64(fma(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) * t_0), sqrt(2.0), 2.0) / fma(1.5, fma(t_2, cos(x), Float64(t_1 * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.0008], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-5], (-N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.00000000000000038e-4

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.3

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

   9. Applied rewrites 62.3%

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

       1. lift--.f64 N/A

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

       2. flip-- N/A

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

       3. metadata-eval N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. rem-square-sqrt N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. metadata-eval N/A

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

       11. add-flip N/A

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

       12. lower--.f64 N/A

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

       13. metadata-eval 62.3

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

   11. Applied rewrites 62.3%

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

   if -8.00000000000000038e-4 < x < 2.50000000000000012e-5

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.0

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

   6. Applied rewrites 62.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. *-commutative N/A

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

      5. div-add-rev N/A

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

      6. +-commutative N/A

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

      7. div-add N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

   8. Applied rewrites 62.1%

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

   if 2.50000000000000012e-5 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.3

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

   9. Applied rewrites 62.3%

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

   11. Applied rewrites 62.3%

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

Alternative 20: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := t\_1 \cdot \cos y\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot t\_0\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, \frac{4}{\sqrt{5} - -1}, \cos y \cdot t\_1\right)}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-6}:\\ \;\;\;\;-\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_2, 1, t\_3\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot t\_0, \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_3\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (* t_1 (cos y))))
   (if (<= x -3.4e-6)
     (/
      (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (pow (sin x) 2.0) t_0))))
      (+
       3.0
       (* 1.5 (fma (cos x) (/ 4.0 (- (sqrt 5.0) -1.0)) (* (cos y) t_1)))))
     (if (<= x 8.8e-6)
       (-
        (/
         (-
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- (cos y) 1.0))))
          2.0)
         (+ 3.0 (* 3.0 (/ (fma t_2 1.0 t_3) 2.0)))))
       (/
        (fma
         (* (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_0)
         (sqrt 2.0)
         2.0)
        (fma 1.5 (fma t_2 (cos x) t_3) 3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = t_1 * cos(y);
	double tmp;
	if (x <= -3.4e-6) {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(x), 2.0) * t_0)))) / (3.0 + (1.5 * fma(cos(x), (4.0 / (sqrt(5.0) - -1.0)), (cos(y) * t_1))));
	} else if (x <= 8.8e-6) {
		tmp = -(((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / (3.0 + (3.0 * (fma(t_2, 1.0, t_3) / 2.0))));
	} else {
		tmp = fma(((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))) * t_0), sqrt(2.0), 2.0) / fma(1.5, fma(t_2, cos(x), t_3), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(t_1 * cos(y))
	tmp = 0.0
	if (x <= -3.4e-6)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(x) ^ 2.0) * t_0)))) / Float64(3.0 + Float64(1.5 * fma(cos(x), Float64(4.0 / Float64(sqrt(5.0) - -1.0)), Float64(cos(y) * t_1)))));
	elseif (x <= 8.8e-6)
		tmp = Float64(-Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_2, 1.0, t_3) / 2.0)))));
	else
		tmp = Float64(fma(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) * t_0), sqrt(2.0), 2.0) / fma(1.5, fma(t_2, cos(x), t_3), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-6], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e-6], (-N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$2 * 1.0 + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.40000000000000006e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.3

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

   9. Applied rewrites 62.3%

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

       1. lift--.f64 N/A

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

       2. flip-- N/A

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

       3. metadata-eval N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. rem-square-sqrt N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. metadata-eval N/A

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

       11. add-flip N/A

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

       12. lower--.f64 N/A

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

       13. metadata-eval 62.3

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

   11. Applied rewrites 62.3%

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

   if -3.40000000000000006e-6 < x < 8.8000000000000004e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.0

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

   6. Applied rewrites 62.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 59.1%

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

   if 8.8000000000000004e-6 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.3

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

   9. Applied rewrites 62.3%

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

   11. Applied rewrites 62.3%

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

Alternative 21: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \left(\cos y - 1\right) \cdot \sqrt{2}\\ t_3 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1 \cdot \cos y\right)\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(t\_2 \cdot t\_0, -0.0625, -2\right)}{3 + 3 \cdot \frac{t\_3}{2}}\\ \mathbf{elif}\;y \leq 0.00044:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\sqrt{5} - -1}, 0.5 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(-0.0625 \cdot t\_0, t\_2, -2\right)}{\mathsf{fma}\left(3, t\_3 \cdot 0.5, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* (- (cos y) 1.0) (sqrt 2.0)))
        (t_3 (fma (- (sqrt 5.0) 1.0) (cos x) (* t_1 (cos y)))))
   (if (<= y -3.3e-6)
     (- (/ (fma (* t_2 t_0) -0.0625 -2.0) (+ 3.0 (* 3.0 (/ t_3 2.0)))))
     (if (<= y 0.00044)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (+
          1.0
          (fma 0.5 (* (cos x) (/ 4.0 (- (sqrt 5.0) -1.0))) (* 0.5 t_1)))))
       (/ (- (fma (* -0.0625 t_0) t_2 -2.0)) (fma 3.0 (* t_3 0.5) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * y)));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (cos(y) - 1.0) * sqrt(2.0);
	double t_3 = fma((sqrt(5.0) - 1.0), cos(x), (t_1 * cos(y)));
	double tmp;
	if (y <= -3.3e-6) {
		tmp = -(fma((t_2 * t_0), -0.0625, -2.0) / (3.0 + (3.0 * (t_3 / 2.0))));
	} else if (y <= 0.00044) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.0 + fma(0.5, (cos(x) * (4.0 / (sqrt(5.0) - -1.0))), (0.5 * t_1))));
	} else {
		tmp = -fma((-0.0625 * t_0), t_2, -2.0) / fma(3.0, (t_3 * 0.5), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(cos(y) - 1.0) * sqrt(2.0))
	t_3 = fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_1 * cos(y)))
	tmp = 0.0
	if (y <= -3.3e-6)
		tmp = Float64(-Float64(fma(Float64(t_2 * t_0), -0.0625, -2.0) / Float64(3.0 + Float64(3.0 * Float64(t_3 / 2.0)))));
	elseif (y <= 0.00044)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(4.0 / Float64(sqrt(5.0) - -1.0))), Float64(0.5 * t_1)))));
	else
		tmp = Float64(Float64(-fma(Float64(-0.0625 * t_0), t_2, -2.0)) / fma(3.0, Float64(t_3 * 0.5), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.3e-6], (-N[(N[(N[(t$95$2 * t$95$0), $MachinePrecision] * -0.0625 + -2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[y, 0.00044], N[(0.3333333333333333 * 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[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(-0.0625 * t$95$0), $MachinePrecision] * t$95$2 + -2.0), $MachinePrecision]) / N[(3.0 * N[(t$95$3 * 0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.30000000000000017e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.0

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

   6. Applied rewrites 62.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 62.0

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

   8. Applied rewrites 62.0%

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

   if -3.30000000000000017e-6 < y < 4.40000000000000016e-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. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 60.0

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\sqrt{5} - -1}, 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 60.0%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\color{blue}{\sqrt{5} - -1}}, 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if 4.40000000000000016e-4 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.0

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

   6. Applied rewrites 62.0%

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

   8. Applied rewrites 62.0%

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

Alternative 22: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_2 := \frac{\mathsf{fma}\left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\cos x - 1\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-6}:\\ \;\;\;\;-\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_0, 1, t\_1\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_2
         (/
          (fma
           (* (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) (- (cos x) 1.0))
           (sqrt 2.0)
           2.0)
          (fma 1.5 (fma t_0 (cos x) t_1) 3.0))))
   (if (<= x -3.4e-6)
     t_2
     (if (<= x 8.8e-6)
       (-
        (/
         (-
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- (cos y) 1.0))))
          2.0)
         (+ 3.0 (* 3.0 (/ (fma t_0 1.0 t_1) 2.0)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (3.0 - sqrt(5.0)) * cos(y);
	double t_2 = fma(((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))) * (cos(x) - 1.0)), sqrt(2.0), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	double tmp;
	if (x <= -3.4e-6) {
		tmp = t_2;
	} else if (x <= 8.8e-6) {
		tmp = -(((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / (3.0 + (3.0 * (fma(t_0, 1.0, t_1) / 2.0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_2 = Float64(fma(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) * Float64(cos(x) - 1.0)), sqrt(2.0), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0))
	tmp = 0.0
	if (x <= -3.4e-6)
		tmp = t_2;
	elseif (x <= 8.8e-6)
		tmp = Float64(-Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_0, 1.0, t_1) / 2.0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-6], t$95$2, If[LessEqual[x, 8.8e-6], (-N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000006e-6 or 8.8000000000000004e-6 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.3

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

   9. Applied rewrites 62.3%

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

   11. Applied rewrites 62.3%

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

   if -3.40000000000000006e-6 < x < 8.8000000000000004e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.0

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

   6. Applied rewrites 62.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 59.1%

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

Alternative 23: 78.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\\ t_2 := 0.5 \cdot t\_0\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t\_1}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\sqrt{5} - -1}, t\_2\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;-\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1, t\_0 \cdot \cos y\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t\_1}{\mathsf{fma}\left(1 + \frac{\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x}{t\_2}, t\_2, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
        (t_2 (* 0.5 t_0)))
   (if (<= x -3.8e-6)
     (*
      0.3333333333333333
      (/ t_1 (+ 1.0 (fma 0.5 (* (cos x) (/ 4.0 (- (sqrt 5.0) -1.0))) t_2))))
     (if (<= x 1.2e-5)
       (-
        (/
         (-
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- (cos y) 1.0))))
          2.0)
         (+ 3.0 (* 3.0 (/ (fma (- (sqrt 5.0) 1.0) 1.0 (* t_0 (cos y))) 2.0)))))
       (*
        0.3333333333333333
        (/
         t_1
         (fma
          (+ 1.0 (/ (* (fma (sqrt 5.0) 0.5 -0.5) (cos x)) t_2))
          t_2
          1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))));
	double t_2 = 0.5 * t_0;
	double tmp;
	if (x <= -3.8e-6) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + fma(0.5, (cos(x) * (4.0 / (sqrt(5.0) - -1.0))), t_2)));
	} else if (x <= 1.2e-5) {
		tmp = -(((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / (3.0 + (3.0 * (fma((sqrt(5.0) - 1.0), 1.0, (t_0 * cos(y))) / 2.0))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / fma((1.0 + ((fma(sqrt(5.0), 0.5, -0.5) * cos(x)) / t_2)), t_2, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))
	t_2 = Float64(0.5 * t_0)
	tmp = 0.0
	if (x <= -3.8e-6)
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(4.0 / Float64(sqrt(5.0) - -1.0))), t_2))));
	elseif (x <= 1.2e-5)
		tmp = Float64(-Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(Float64(sqrt(5.0) - 1.0), 1.0, Float64(t_0 * cos(y))) / 2.0)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_1 / fma(Float64(1.0 + Float64(Float64(fma(sqrt(5.0), 0.5, -0.5) * cos(x)) / t_2)), t_2, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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]}, Block[{t$95$2 = N[(0.5 * t$95$0), $MachinePrecision]}, If[LessEqual[x, -3.8e-6], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-5], (-N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * 1.0 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(0.3333333333333333 * N[(t$95$1 / N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e-6

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 60.0

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\sqrt{5} - -1}, 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 60.0%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\color{blue}{\sqrt{5} - -1}}, 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -3.8e-6 < x < 1.2e-5

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.0

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

   6. Applied rewrites 62.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 59.1%

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

   if 1.2e-5 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{1}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1} \]

      4. +-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1} \]

      5. sum-to-mult N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(1 + \frac{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(1 + \frac{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, \color{blue}{\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 1\right)} \]

   6. Applied rewrites 59.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(1 + \frac{\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x}{0.5 \cdot \left(3 - \sqrt{5}\right)}, \color{blue}{0.5 \cdot \left(3 - \sqrt{5}\right)}, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\sqrt{5} - -1}, 0.5 \cdot t\_0\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;-\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_2, 1, t\_0 \cdot \cos y\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, \left(t\_1 \cdot \sqrt{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), -2\right)}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_2, \cos x, t\_0\right), -1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) 1.0))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -3.8e-6)
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_1))))
       (+ 1.0 (fma 0.5 (* (cos x) (/ 4.0 (- (sqrt 5.0) -1.0))) (* 0.5 t_0)))))
     (if (<= x 1.2e-5)
       (-
        (/
         (-
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- (cos y) 1.0))))
          2.0)
         (+ 3.0 (* 3.0 (/ (fma t_2 1.0 (* t_0 (cos y))) 2.0)))))
       (*
        (/
         (fma
          0.0625
          (* (* t_1 (sqrt 2.0)) (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          -2.0)
         (fma -0.5 (fma t_2 (cos x) t_0) -1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -3.8e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_1)))) / (1.0 + fma(0.5, (cos(x) * (4.0 / (sqrt(5.0) - -1.0))), (0.5 * t_0))));
	} else if (x <= 1.2e-5) {
		tmp = -(((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / (3.0 + (3.0 * (fma(t_2, 1.0, (t_0 * cos(y))) / 2.0))));
	} else {
		tmp = (fma(0.0625, ((t_1 * sqrt(2.0)) * (0.5 - (0.5 * cos((2.0 * x))))), -2.0) / fma(-0.5, fma(t_2, cos(x), t_0), -1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -3.8e-6)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_1)))) / Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(4.0 / Float64(sqrt(5.0) - -1.0))), Float64(0.5 * t_0)))));
	elseif (x <= 1.2e-5)
		tmp = Float64(-Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_2, 1.0, Float64(t_0 * cos(y))) / 2.0)))));
	else
		tmp = Float64(Float64(fma(0.0625, Float64(Float64(t_1 * sqrt(2.0)) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), -2.0) / fma(-0.5, fma(t_2, cos(x), t_0), -1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -3.8e-6], N[(0.3333333333333333 * 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[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-5], (-N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$2 * 1.0 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(N[(0.0625 * N[(N[(t$95$1 * 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[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e-6

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - \color{blue}{1}\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\color{blue}{\sqrt{5} + 1}}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \frac{\sqrt{5} \cdot \sqrt{5} - 1}{\sqrt{5} + 1}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \frac{\sqrt{5} \cdot \sqrt{5} - 1}{\sqrt{5} + 1}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \frac{\sqrt{5} \cdot \sqrt{5} - 1}{\sqrt{5} + 1}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. rem-square-sqrt N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \frac{5 - 1}{\sqrt{\color{blue}{5}} + 1}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \frac{4}{\color{blue}{\sqrt{5}} + 1}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \frac{2 \cdot 2}{\color{blue}{\sqrt{5}} + 1}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \frac{2 \cdot 2}{\color{blue}{\sqrt{5} + 1}}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \frac{4}{\color{blue}{\sqrt{5}} + 1}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \frac{4}{\sqrt{5} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \frac{4}{\sqrt{5} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. metadata-eval 60.0

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\sqrt{5} - -1}, 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 60.0%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \frac{4}{\color{blue}{\sqrt{5} - -1}}, 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -3.8e-6 < x < 1.2e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Taylor expanded in x around 0

      \[\leadsto -\frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - \color{blue}{2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      4. lower-pow.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      5. lower-sin.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      8. lower--.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      9. lower-cos.f64 62.0

         \[\leadsto -\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 62.0%

      \[\leadsto -\frac{\color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   7. Taylor expanded in x around 0

      \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.1%

      \[\leadsto -\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   if 1.2e-5 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), -2\right)}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), -2\right)\\ t_2 := \sqrt{5} - 1\\ t_3 := \mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_2, \cos x, t\_0\right), -1\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{t\_3}{t\_1}}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;-\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_2, 1, t\_0 \cdot \cos y\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_3} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (fma
          0.0625
          (* (* (- (cos x) 1.0) (sqrt 2.0)) (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          -2.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (fma -0.5 (fma t_2 (cos x) t_0) -1.0)))
   (if (<= x -3.8e-6)
     (/ 0.3333333333333333 (/ t_3 t_1))
     (if (<= x 1.2e-5)
       (-
        (/
         (-
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- (cos y) 1.0))))
          2.0)
         (+ 3.0 (* 3.0 (/ (fma t_2 1.0 (* t_0 (cos y))) 2.0)))))
       (* (/ t_1 t_3) 0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(0.0625, (((cos(x) - 1.0) * sqrt(2.0)) * (0.5 - (0.5 * cos((2.0 * x))))), -2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma(-0.5, fma(t_2, cos(x), t_0), -1.0);
	double tmp;
	if (x <= -3.8e-6) {
		tmp = 0.3333333333333333 / (t_3 / t_1);
	} else if (x <= 1.2e-5) {
		tmp = -(((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / (3.0 + (3.0 * (fma(t_2, 1.0, (t_0 * cos(y))) / 2.0))));
	} else {
		tmp = (t_1 / t_3) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = 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)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = fma(-0.5, fma(t_2, cos(x), t_0), -1.0)
	tmp = 0.0
	if (x <= -3.8e-6)
		tmp = Float64(0.3333333333333333 / Float64(t_3 / t_1));
	elseif (x <= 1.2e-5)
		tmp = Float64(-Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_2, 1.0, Float64(t_0 * cos(y))) / 2.0)))));
	else
		tmp = Float64(Float64(t_1 / t_3) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -3.8e-6], N[(0.3333333333333333 / N[(t$95$3 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-5], (-N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$2 * 1.0 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(t$95$1 / t$95$3), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e-6

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 60.0%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}{\mathsf{fma}\left(0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), -2\right)}}} \]

   if -3.8e-6 < x < 1.2e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Taylor expanded in x around 0

      \[\leadsto -\frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - \color{blue}{2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      4. lower-pow.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      5. lower-sin.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      8. lower--.f64 N/A

         \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

      9. lower-cos.f64 62.0

         \[\leadsto -\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   6. Applied rewrites 62.0%

      \[\leadsto -\frac{\color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   7. Taylor expanded in x around 0

      \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.1%

      \[\leadsto -\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   if 1.2e-5 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), -2\right)}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), -2\right)\\ t_2 := \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right), -1\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{t\_2}{t\_1}}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_0\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (fma
          0.0625
          (* (* (- (cos x) 1.0) (sqrt 2.0)) (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          -2.0))
        (t_2 (fma -0.5 (fma (- (sqrt 5.0) 1.0) (cos x) t_0) -1.0)))
   (if (<= x -3.8e-6)
     (/ 0.3333333333333333 (/ t_2 t_1))
     (if (<= x 1.2e-5)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* (cos y) t_0)) 1.0))))
       (* (/ t_1 t_2) 0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(0.0625, (((cos(x) - 1.0) * sqrt(2.0)) * (0.5 - (0.5 * cos((2.0 * x))))), -2.0);
	double t_2 = fma(-0.5, fma((sqrt(5.0) - 1.0), cos(x), t_0), -1.0);
	double tmp;
	if (x <= -3.8e-6) {
		tmp = 0.3333333333333333 / (t_2 / t_1);
	} else if (x <= 1.2e-5) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + (cos(y) * t_0)) - 1.0)));
	} else {
		tmp = (t_1 / t_2) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = 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)
	t_2 = fma(-0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), t_0), -1.0)
	tmp = 0.0
	if (x <= -3.8e-6)
		tmp = Float64(0.3333333333333333 / Float64(t_2 / t_1));
	elseif (x <= 1.2e-5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * t_0)) - 1.0))));
	else
		tmp = Float64(Float64(t_1 / t_2) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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]}, Block[{t$95$2 = N[(-0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -3.8e-6], N[(0.3333333333333333 / N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-5], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e-6

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 60.0%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}{\mathsf{fma}\left(0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), -2\right)}}} \]

   if -3.8e-6 < x < 1.2e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}} \]

   6. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}} \]

   if 1.2e-5 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 60.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), -2\right)}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)} \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 60.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), -2\right)}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    0.0625
    (* (* (- (cos x) 1.0) (sqrt 2.0)) (- 0.5 (* 0.5 (cos (* 2.0 x)))))
    -2.0)
   (fma -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 60.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

6. Applied rewrites 60.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), -2\right)}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)} \cdot 0.3333333333333333} \]
7. Add Preprocessing

Alternative 28: 45.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -\frac{-2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (-
  (/
   -2.0
   (+
    3.0
    (*
     3.0
     (/
      (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
      2.0))))))

C

double code(double x, double y) {
	return -(-2.0 / (3.0 + (3.0 * (fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))) / 2.0))));
}

Julia

function code(x, y)
	return Float64(-Float64(-2.0 / Float64(3.0 + Float64(3.0 * Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) / 2.0)))))
end

Wolfram

code[x_, y_] := (-N[(-2.0 / N[(3.0 + N[(3.0 * N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}} \]

4. Taylor expanded in x around 0

   \[\leadsto -\frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]
5. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - \color{blue}{2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   4. lower-pow.f64 N/A

      \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   5. lower-sin.f64 N/A

      \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   8. lower--.f64 N/A

      \[\leadsto -\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

   9. lower-cos.f64 62.0

      \[\leadsto -\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

6. Applied rewrites 62.0%

   \[\leadsto -\frac{\color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]

7. Taylor expanded in y around 0

   \[\leadsto -\frac{-2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]
8. Step-by-step derivation

9. Applied rewrites 45.1%

   \[\leadsto -\frac{-2}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \]
10. Add Preprocessing

Alternative 29: 42.8% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   2.0
   (+
    1.0
    (fma 0.5 (* (cos x) (- (sqrt 5.0) 1.0)) (* 0.5 (- 3.0 (sqrt 5.0))))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * (2.0 / (1.0 + fma(0.5, (cos(x) * (sqrt(5.0) - 1.0)), (0.5 * (3.0 - sqrt(5.0))))));
}

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(2.0 / Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(sqrt(5.0) - 1.0)), Float64(0.5 * Float64(3.0 - sqrt(5.0)))))))
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(2.0 / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 60.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \cdot \frac{2}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 42.8%

   \[\leadsto 0.3333333333333333 \cdot \frac{2}{\color{blue}{1} + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
8. Add Preprocessing

Alternative 30: 40.2% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(1 - \sqrt{5}\right), 0.5, 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (fma (- (- 3.0 (sqrt 5.0)) (- 1.0 (sqrt 5.0))) 0.5 1.0)))

C

double code(double x, double y) {
	return 0.6666666666666666 / fma(((3.0 - sqrt(5.0)) - (1.0 - sqrt(5.0))), 0.5, 1.0);
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 / fma(Float64(Float64(3.0 - sqrt(5.0)) - Float64(1.0 - sqrt(5.0))), 0.5, 1.0))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] - N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 60.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{2}{3}}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \color{blue}{\sqrt{5}}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. lower-sqrt.f64 40.2

      \[\leadsto \frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, 3 - \sqrt{5}, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)} \]

7. Applied rewrites 40.2%

   \[\leadsto \frac{0.6666666666666666}{\color{blue}{1 + \mathsf{fma}\left(0.5, 3 - \sqrt{5}, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{3 - \sqrt{5}}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, 3 - \sqrt{5}, \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1} \]

   3. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1} \]

   5. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 1} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \left(3 - \sqrt{5}\right) + \color{blue}{\left(\sqrt{5} - 1\right)}, 1\right)} \]

   7. lower-+.f64 40.2

      \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - \color{blue}{1}\right), 1\right)} \]

9. Applied rewrites 40.2%

   \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \left(3 - \sqrt{5}\right) + \color{blue}{\left(\sqrt{5} - 1\right)}, 1\right)} \]
10. Step-by-step derivation

    1. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 1} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{2}{3}}{\left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + 1} \]

    3. lower-fma.f64 40.2

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 0.5, 1\right)} \]

    4. lift-+.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), \frac{1}{2}, 1\right)} \]

    5. add-flip N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(\mathsf{neg}\left(\left(\sqrt{5} - 1\right)\right)\right), \frac{1}{2}, 1\right)} \]

    6. lower--.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(\mathsf{neg}\left(\left(\sqrt{5} - 1\right)\right)\right), \frac{1}{2}, 1\right)} \]

    7. lift--.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(\mathsf{neg}\left(\left(\sqrt{5} - 1\right)\right)\right), \frac{1}{2}, 1\right)} \]

    8. sub-negate-rev N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(1 - \sqrt{5}\right), \frac{1}{2}, 1\right)} \]

    9. lower--.f64 40.2

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(1 - \sqrt{5}\right), 0.5, 1\right)} \]

11. Applied rewrites 40.2%

    \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) - \left(1 - \sqrt{5}\right), \color{blue}{0.5}, 1\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (- (sin y) (/ (sin x) 16.0))) (- (cos x) (cos y)))) (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))