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

Percentage Accurate: 99.3% → 99.4%

Time: 12.5s

Alternatives: 28

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.1× speedup

Math

\[\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.6180339887498949, \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (- (cos x) (cos y)) (sqrt 2.0))
   (* (fma (sin y) -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y)))
   2.0)
  (fma
   (fma 0.6180339887498949 (cos x) 1.0)
   3.0
   (* 1.1458980337503155 (cos y)))))

C

double code(double x, double y) {
	return fma(((cos(x) - cos(y)) * sqrt(2.0)), (fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(fma(0.6180339887498949, cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(fma(0.6180339887498949, cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.6180339887498949 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $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. Evaluated real constant 99.3%

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

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

6. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

7. Evaluated real constant 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.6180339887498949}, \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 1.1× speedup

Math

\[\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.6180339887498949, \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (* (- (cos x) (cos y)) (sqrt 2.0)) (fma -0.0625 (sin y) (sin x)))
   (fma -0.0625 (sin x) (sin y))
   2.0)
  (fma
   (fma 0.6180339887498949 (cos x) 1.0)
   3.0
   (* 1.1458980337503155 (cos y)))))

C

double code(double x, double y) {
	return fma((((cos(x) - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(fma(0.6180339887498949, cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(fma(0.6180339887498949, cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.6180339887498949 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $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. Evaluated real constant 99.3%

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

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

5. Evaluated real constant 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.6180339887498949}, \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
6. Add Preprocessing

Alternative 3: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)\\ t_1 := \cos x - \cos y\\ t_2 := t\_1 \cdot \sqrt{2}\\ t_3 := y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\\ \mathbf{if}\;y \leq -0.031:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.0205:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \mathsf{fma}\left(-0.0625, t\_3, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, t\_3\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sin y, 2\right)}{t\_0}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
          3.0
          (* 1.1458980337503155 (cos y))))
        (t_1 (- (cos x) (cos y)))
        (t_2 (* t_1 (sqrt 2.0)))
        (t_3 (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0))))))
   (if (<= y -0.031)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_1))
      (*
       3.0
       (+
        (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
        (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
     (if (<= y 0.0205)
       (/
        (fma (* t_2 (fma -0.0625 t_3 (sin x))) (fma -0.0625 (sin x) t_3) 2.0)
        t_0)
       (/ (fma (* t_2 (fma -0.0625 (sin y) (sin x))) (sin y) 2.0) t_0)))))

C

double code(double x, double y) {
	double t_0 = fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	double t_1 = cos(x) - cos(y);
	double t_2 = t_1 * sqrt(2.0);
	double t_3 = y * (1.0 + (-0.16666666666666666 * pow(y, 2.0)));
	double tmp;
	if (y <= -0.031) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_1)) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	} else if (y <= 0.0205) {
		tmp = fma((t_2 * fma(-0.0625, t_3, sin(x))), fma(-0.0625, sin(x), t_3), 2.0) / t_0;
	} else {
		tmp = fma((t_2 * fma(-0.0625, sin(y), sin(x))), sin(y), 2.0) / t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y)))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(t_1 * sqrt(2.0))
	t_3 = Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0))))
	tmp = 0.0
	if (y <= -0.031)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_1)) / 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)))));
	elseif (y <= 0.0205)
		tmp = Float64(fma(Float64(t_2 * fma(-0.0625, t_3, sin(x))), fma(-0.0625, sin(x), t_3), 2.0) / t_0);
	else
		tmp = Float64(fma(Float64(t_2 * fma(-0.0625, sin(y), sin(x))), sin(y), 2.0) / t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.031], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $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], If[LessEqual[y, 0.0205], N[(N[(N[(t$95$2 * N[(-0.0625 * t$95$3 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + t$95$3), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(t$95$2 * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.031

   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 64.0%

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

   4. Applied rewrites 64.0%

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

   if -0.031 < y < 0.0205000000000000009

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, y \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}, \sin x\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, y \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {y}^{2}}\right), \sin x\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, y \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right), \sin x\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-pow.f64 50.7%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{\color{blue}{2}}\right), \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Applied rewrites 50.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \color{blue}{y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right), \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right), \sin x\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, y \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right), \sin x\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, y \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {y}^{2}}\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right), \sin x\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, y \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{y}^{2}}\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-pow.f64 50.2%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right), \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{\color{blue}{2}}\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   10. Applied rewrites 50.2%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right), \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \color{blue}{y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   if 0.0205000000000000009 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-sin.f64 64.0%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Applied rewrites 64.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ \mathbf{if}\;y \leq -0.031:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.0205:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - t\_1, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(t\_1, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_0 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y))) (t_1 (fma (* y y) -0.5 1.0)))
   (if (<= y -0.031)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0))
      (*
       3.0
       (+
        (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
        (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
     (if (<= y 0.0205)
       (/
        (fma
         (- (cos x) t_1)
         (*
          (fma (sin x) -0.0625 (sin y))
          (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
         2.0)
        (*
         (fma
          t_1
          0.38196601125010515
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
         3.0))
       (/
        (fma (* (* t_0 (sqrt 2.0)) (fma -0.0625 (sin y) (sin x))) (sin y) 2.0)
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
         3.0
         (* 1.1458980337503155 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma((y * y), -0.5, 1.0);
	double tmp;
	if (y <= -0.031) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	} else if (y <= 0.0205) {
		tmp = fma((cos(x) - t_1), (fma(sin(x), -0.0625, sin(y)) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / (fma(t_1, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = fma(((t_0 * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), sin(y), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(Float64(y * y), -0.5, 1.0)
	tmp = 0.0
	if (y <= -0.031)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	elseif (y <= 0.0205)
		tmp = Float64(fma(Float64(cos(x) - t_1), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / Float64(fma(t_1, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = Float64(fma(Float64(Float64(t_0 * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), sin(y), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[y, -0.031], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $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], If[LessEqual[y, 0.0205], N[(N[(N[(N[Cos[x], $MachinePrecision] - t$95$1), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $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[(t$95$1 * 0.38196601125010515 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.031

   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 64.0%

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

   4. Applied rewrites 64.0%

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

   if -0.031 < y < 0.0205000000000000009

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.3%

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.6%

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

   8. Applied rewrites 52.6%

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

   10. Applied rewrites 52.6%

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

   if 0.0205000000000000009 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-sin.f64 64.0%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Applied rewrites 64.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ \mathbf{if}\;y \leq -0.031:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.0205:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - t\_1, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(t\_1, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_0 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y))) (t_1 (fma (* y y) -0.5 1.0)))
   (if (<= y -0.031)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0))
      (*
       3.0
       (+
        (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
        (* 0.38196601125010515 (cos y)))))
     (if (<= y 0.0205)
       (/
        (fma
         (- (cos x) t_1)
         (*
          (fma (sin x) -0.0625 (sin y))
          (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
         2.0)
        (*
         (fma
          t_1
          0.38196601125010515
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
         3.0))
       (/
        (fma (* (* t_0 (sqrt 2.0)) (fma -0.0625 (sin y) (sin x))) (sin y) 2.0)
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
         3.0
         (* 1.1458980337503155 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma((y * y), -0.5, 1.0);
	double tmp;
	if (y <= -0.031) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (0.38196601125010515 * cos(y))));
	} else if (y <= 0.0205) {
		tmp = fma((cos(x) - t_1), (fma(sin(x), -0.0625, sin(y)) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / (fma(t_1, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = fma(((t_0 * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), sin(y), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(Float64(y * y), -0.5, 1.0)
	tmp = 0.0
	if (y <= -0.031)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(0.38196601125010515 * cos(y)))));
	elseif (y <= 0.0205)
		tmp = Float64(fma(Float64(cos(x) - t_1), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / Float64(fma(t_1, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = Float64(fma(Float64(Float64(t_0 * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), sin(y), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[y, -0.031], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $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[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0205], N[(N[(N[(N[Cos[x], $MachinePrecision] - t$95$1), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $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[(t$95$1 * 0.38196601125010515 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.031

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.0%

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

   5. Applied rewrites 64.0%

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

   if -0.031 < y < 0.0205000000000000009

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.3%

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.6%

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

   8. Applied rewrites 52.6%

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

   10. Applied rewrites 52.6%

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

   if 0.0205000000000000009 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-sin.f64 64.0%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Applied rewrites 64.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)\\ t_2 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_3 := \left(\cos x - \cos y\right) \cdot \sqrt{2}\\ \mathbf{if}\;y \leq -0.031:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, t\_0 \cdot \sin y, 2\right)}{t\_1}\\ \mathbf{elif}\;y \leq 0.0205:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - t\_2, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(t\_0 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(t\_2, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3 \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sin y, 2\right)}{t\_1}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1
         (fma
          (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
          3.0
          (* 1.1458980337503155 (cos y))))
        (t_2 (fma (* y y) -0.5 1.0))
        (t_3 (* (- (cos x) (cos y)) (sqrt 2.0))))
   (if (<= y -0.031)
     (/ (fma t_3 (* t_0 (sin y)) 2.0) t_1)
     (if (<= y 0.0205)
       (/
        (fma
         (- (cos x) t_2)
         (* (fma (sin x) -0.0625 (sin y)) (* t_0 (sqrt 2.0)))
         2.0)
        (*
         (fma
          t_2
          0.38196601125010515
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
         3.0))
       (/ (fma (* t_3 (fma -0.0625 (sin y) (sin x))) (sin y) 2.0) t_1)))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	double t_2 = fma((y * y), -0.5, 1.0);
	double t_3 = (cos(x) - cos(y)) * sqrt(2.0);
	double tmp;
	if (y <= -0.031) {
		tmp = fma(t_3, (t_0 * sin(y)), 2.0) / t_1;
	} else if (y <= 0.0205) {
		tmp = fma((cos(x) - t_2), (fma(sin(x), -0.0625, sin(y)) * (t_0 * sqrt(2.0))), 2.0) / (fma(t_2, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = fma((t_3 * fma(-0.0625, sin(y), sin(x))), sin(y), 2.0) / t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y)))
	t_2 = fma(Float64(y * y), -0.5, 1.0)
	t_3 = Float64(Float64(cos(x) - cos(y)) * sqrt(2.0))
	tmp = 0.0
	if (y <= -0.031)
		tmp = Float64(fma(t_3, Float64(t_0 * sin(y)), 2.0) / t_1);
	elseif (y <= 0.0205)
		tmp = Float64(fma(Float64(cos(x) - t_2), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(t_0 * sqrt(2.0))), 2.0) / Float64(fma(t_2, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = Float64(fma(Float64(t_3 * fma(-0.0625, sin(y), sin(x))), sin(y), 2.0) / t_1);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.031], N[(N[(t$95$3 * N[(t$95$0 * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 0.0205], N[(N[(N[(N[Cos[x], $MachinePrecision] - t$95$2), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$2 * 0.38196601125010515 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$3 * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.031

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   8. Step-by-step derivation

      1. lower-sin.f64 64.0%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   9. Applied rewrites 64.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   if -0.031 < y < 0.0205000000000000009

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.3%

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.6%

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

   8. Applied rewrites 52.6%

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

   10. Applied rewrites 52.6%

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

   if 0.0205000000000000009 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-sin.f64 64.0%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Applied rewrites 64.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_1 := \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.031:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0205:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - t\_0, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(t\_0, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) -0.5 1.0))
        (t_1
         (/
          (fma
           (* (* (- (cos x) (cos y)) (sqrt 2.0)) (fma -0.0625 (sin y) (sin x)))
           (sin y)
           2.0)
          (fma
           (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
           3.0
           (* 1.1458980337503155 (cos y))))))
   (if (<= y -0.031)
     t_1
     (if (<= y 0.0205)
       (/
        (fma
         (- (cos x) t_0)
         (*
          (fma (sin x) -0.0625 (sin y))
          (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
         2.0)
        (*
         (fma
          t_0
          0.38196601125010515
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
         3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), -0.5, 1.0);
	double t_1 = fma((((cos(x) - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), sin(y), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	double tmp;
	if (y <= -0.031) {
		tmp = t_1;
	} else if (y <= 0.0205) {
		tmp = fma((cos(x) - t_0), (fma(sin(x), -0.0625, sin(y)) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / (fma(t_0, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), -0.5, 1.0)
	t_1 = Float64(fma(Float64(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), sin(y), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))))
	tmp = 0.0
	if (y <= -0.031)
		tmp = t_1;
	elseif (y <= 0.0205)
		tmp = Float64(fma(Float64(cos(x) - t_0), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), 2.0) / Float64(fma(t_0, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.031], t$95$1, If[LessEqual[y, 0.0205], N[(N[(N[(N[Cos[x], $MachinePrecision] - t$95$0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $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[(t$95$0 * 0.38196601125010515 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.031 or 0.0205000000000000009 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-sin.f64 64.0%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Applied rewrites 64.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   if -0.031 < y < 0.0205000000000000009

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.3%

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.6%

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

   8. Applied rewrites 52.6%

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

   10. Applied rewrites 52.6%

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

Alternative 8: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_1 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_2 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.049:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_2 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.6180339887498949, \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.047:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - t\_0, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(t\_1 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(t\_0, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \left(\sin y \cdot t\_2\right), 1\right) + 1}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) -0.5 1.0))
        (t_1 (fma (sin y) -0.0625 (sin x)))
        (t_2 (- 1.0 (cos y))))
   (if (<= y -0.049)
     (/
      (fma
       (* (* t_2 (sqrt 2.0)) (fma -0.0625 (sin y) (sin x)))
       (fma -0.0625 (sin x) (sin y))
       2.0)
      (fma
       (fma 0.6180339887498949 (cos x) 1.0)
       3.0
       (* 1.1458980337503155 (cos y))))
     (if (<= y 0.047)
       (/
        (fma
         (- (cos x) t_0)
         (* (fma (sin x) -0.0625 (sin y)) (* t_1 (sqrt 2.0)))
         2.0)
        (*
         (fma
          t_0
          0.38196601125010515
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
         3.0))
       (/
        (+ (fma t_1 (* (sqrt 2.0) (* (sin y) t_2)) 1.0) 1.0)
        (*
         3.0
         (+
          (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
          (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), -0.5, 1.0);
	double t_1 = fma(sin(y), -0.0625, sin(x));
	double t_2 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.049) {
		tmp = fma(((t_2 * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(fma(0.6180339887498949, cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	} else if (y <= 0.047) {
		tmp = fma((cos(x) - t_0), (fma(sin(x), -0.0625, sin(y)) * (t_1 * sqrt(2.0))), 2.0) / (fma(t_0, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = (fma(t_1, (sqrt(2.0) * (sin(y) * t_2)), 1.0) + 1.0) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), -0.5, 1.0)
	t_1 = fma(sin(y), -0.0625, sin(x))
	t_2 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.049)
		tmp = Float64(fma(Float64(Float64(t_2 * sqrt(2.0)) * fma(-0.0625, sin(y), sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(fma(0.6180339887498949, cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))));
	elseif (y <= 0.047)
		tmp = Float64(fma(Float64(cos(x) - t_0), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(t_1 * sqrt(2.0))), 2.0) / Float64(fma(t_0, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = Float64(Float64(fma(t_1, Float64(sqrt(2.0) * Float64(sin(y) * t_2)), 1.0) + 1.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.049], N[(N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.6180339887498949 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.047], N[(N[(N[(N[Cos[x], $MachinePrecision] - t$95$0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * 0.38196601125010515 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $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]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.049000000000000002

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-cos.f64 62.5%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Applied rewrites 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   8. Evaluated real constant 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.6180339887498949}, \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   if -0.049000000000000002 < y < 0.047

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.3%

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.6%

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

   8. Applied rewrites 52.6%

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

   10. Applied rewrites 52.6%

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

   if 0.047 < y

   1. Initial program 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 62.3%

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

   6. Applied rewrites 62.3%

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

Alternative 9: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_1 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_2 := \frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 1\right) + 1}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.049:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.047:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - t\_0, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(t\_1 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(t\_0, 0.38196601125010515, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) -0.5 1.0))
        (t_1 (fma (sin y) -0.0625 (sin x)))
        (t_2
         (/
          (+ (fma t_1 (* (sqrt 2.0) (* (sin y) (- 1.0 (cos y)))) 1.0) 1.0)
          (*
           3.0
           (+
            (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
            (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))))
   (if (<= y -0.049)
     t_2
     (if (<= y 0.047)
       (/
        (fma
         (- (cos x) t_0)
         (* (fma (sin x) -0.0625 (sin y)) (* t_1 (sqrt 2.0)))
         2.0)
        (*
         (fma
          t_0
          0.38196601125010515
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
         3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), -0.5, 1.0);
	double t_1 = fma(sin(y), -0.0625, sin(x));
	double t_2 = (fma(t_1, (sqrt(2.0) * (sin(y) * (1.0 - cos(y)))), 1.0) + 1.0) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	double tmp;
	if (y <= -0.049) {
		tmp = t_2;
	} else if (y <= 0.047) {
		tmp = fma((cos(x) - t_0), (fma(sin(x), -0.0625, sin(y)) * (t_1 * sqrt(2.0))), 2.0) / (fma(t_0, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), -0.5, 1.0)
	t_1 = fma(sin(y), -0.0625, sin(x))
	t_2 = Float64(Float64(fma(t_1, Float64(sqrt(2.0) * Float64(sin(y) * Float64(1.0 - cos(y)))), 1.0) + 1.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -0.049)
		tmp = t_2;
	elseif (y <= 0.047)
		tmp = Float64(fma(Float64(cos(x) - t_0), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(t_1 * sqrt(2.0))), 2.0) / Float64(fma(t_0, 0.38196601125010515, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $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]}, If[LessEqual[y, -0.049], t$95$2, If[LessEqual[y, 0.047], N[(N[(N[(N[Cos[x], $MachinePrecision] - t$95$0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * 0.38196601125010515 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.049000000000000002 or 0.047 < y

   1. Initial program 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 62.3%

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

   6. Applied rewrites 62.3%

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

   if -0.049000000000000002 < y < 0.047

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.3%

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.6%

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

   8. Applied rewrites 52.6%

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

   10. Applied rewrites 52.6%

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

Alternative 10: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := 1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 1\right) + 1}{3 \cdot \left(t\_0 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ t_2 := 1 + -0.5 \cdot {y}^{2}\\ \mathbf{if}\;y \leq -0.049:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.047:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right) - 0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - t\_2\right)}{3 \cdot \left(t\_0 + 0.38196601125010515 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))))
        (t_1
         (/
          (+
           (fma
            (fma (sin y) -0.0625 (sin x))
            (* (sqrt 2.0) (* (sin y) (- 1.0 (cos y))))
            1.0)
           1.0)
          (* 3.0 (+ t_0 (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))
        (t_2 (+ 1.0 (* -0.5 (pow y 2.0)))))
   (if (<= y -0.049)
     t_1
     (if (<= y 0.047)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ y 16.0)))
           (-
            (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0))))
            (* 0.0625 (sin x))))
          (- (cos x) t_2)))
        (* 3.0 (+ t_0 (* 0.38196601125010515 t_2))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x));
	double t_1 = (fma(fma(sin(y), -0.0625, sin(x)), (sqrt(2.0) * (sin(y) * (1.0 - cos(y)))), 1.0) + 1.0) / (3.0 * (t_0 + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	double t_2 = 1.0 + (-0.5 * pow(y, 2.0));
	double tmp;
	if (y <= -0.049) {
		tmp = t_1;
	} else if (y <= 0.047) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * ((y * (1.0 + (-0.16666666666666666 * pow(y, 2.0)))) - (0.0625 * sin(x)))) * (cos(x) - t_2))) / (3.0 * (t_0 + (0.38196601125010515 * t_2)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)))
	t_1 = Float64(Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sqrt(2.0) * Float64(sin(y) * Float64(1.0 - cos(y)))), 1.0) + 1.0) / Float64(3.0 * Float64(t_0 + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
	t_2 = Float64(1.0 + Float64(-0.5 * (y ^ 2.0)))
	tmp = 0.0
	if (y <= -0.049)
		tmp = t_1;
	elseif (y <= 0.047)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0)))) - Float64(0.0625 * sin(x)))) * Float64(cos(x) - t_2))) / Float64(3.0 * Float64(t_0 + Float64(0.38196601125010515 * t_2))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(-0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.049], t$95$1, If[LessEqual[y, 0.047], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(0.38196601125010515 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.049000000000000002 or 0.047 < y

   1. Initial program 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 62.3%

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

   6. Applied rewrites 62.3%

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

   if -0.049000000000000002 < y < 0.047

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.2%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 49.7%

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

   11. Applied rewrites 49.7%

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

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 50.6%

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

   14. Applied rewrites 50.6%

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

   15. Taylor expanded in y around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot {y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \left(1 + -0.5 \cdot {y}^{2}\right)\right)} \]
   16. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sin.f64 49.5%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right) - 0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot {y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \left(1 + -0.5 \cdot {y}^{2}\right)\right)} \]

   17. Applied rewrites 49.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right) - 0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot {y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \left(1 + -0.5 \cdot {y}^{2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 1\right) + 1}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.049:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.047:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, y\right) \cdot \mathsf{fma}\left(y, -0.0625, \sin x\right), 2\right) \cdot 0.3333333333333333}{\frac{\mathsf{fma}\left(t\_1, \cos x, t\_0 \cdot \cos y\right)}{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (+
           (fma
            (fma (sin y) -0.0625 (sin x))
            (* (sqrt 2.0) (* (sin y) (- 1.0 (cos y))))
            1.0)
           1.0)
          (*
           3.0
           (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))))
   (if (<= y -0.049)
     t_2
     (if (<= y 0.047)
       (/
        (*
         (fma
          (* (- (cos x) (cos y)) (sqrt 2.0))
          (* (fma (sin x) -0.0625 y) (fma y -0.0625 (sin x)))
          2.0)
         0.3333333333333333)
        (+ (/ (fma t_1 (cos x) (* t_0 (cos y))) 2.0) 1.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (fma(fma(sin(y), -0.0625, sin(x)), (sqrt(2.0) * (sin(y) * (1.0 - cos(y)))), 1.0) + 1.0) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	double tmp;
	if (y <= -0.049) {
		tmp = t_2;
	} else if (y <= 0.047) {
		tmp = (fma(((cos(x) - cos(y)) * sqrt(2.0)), (fma(sin(x), -0.0625, y) * fma(y, -0.0625, sin(x))), 2.0) * 0.3333333333333333) / ((fma(t_1, cos(x), (t_0 * cos(y))) / 2.0) + 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sqrt(2.0) * Float64(sin(y) * Float64(1.0 - cos(y)))), 1.0) + 1.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -0.049)
		tmp = t_2;
	elseif (y <= 0.047)
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, y) * fma(y, -0.0625, sin(x))), 2.0) * 0.3333333333333333) / Float64(Float64(fma(t_1, cos(x), Float64(t_0 * cos(y))) / 2.0) + 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.049], t$95$2, If[LessEqual[y, 0.047], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + y), $MachinePrecision] * N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.049000000000000002 or 0.047 < y

   1. Initial program 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 62.3%

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

   6. Applied rewrites 62.3%

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

   if -0.049000000000000002 < y < 0.047

   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 99.2%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 50.2%

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

Alternative 12: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := t\_1 \cdot \sqrt{2}\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.95:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_1}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos y \cdot t\_3, 0.5 \cdot t\_0\right)\right)}\\ \mathbf{elif}\;y \leq 0.047:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \mathsf{fma}\left(\sin x, -0.0625, y\right) \cdot \mathsf{fma}\left(y, -0.0625, \sin x\right), 2\right) \cdot 0.3333333333333333}{\frac{\mathsf{fma}\left(t\_0, \cos x, t\_3 \cdot \cos y\right)}{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (* t_1 (sqrt 2.0)))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= y -0.95)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_1))
      (* 3.0 (+ 1.0 (fma 0.5 (* (cos y) t_3) (* 0.5 t_0)))))
     (if (<= y 0.047)
       (/
        (*
         (fma t_2 (* (fma (sin x) -0.0625 y) (fma y -0.0625 (sin x))) 2.0)
         0.3333333333333333)
        (+ (/ (fma t_0 (cos x) (* t_3 (cos y))) 2.0) 1.0))
       (/
        (fma t_2 (* -0.0625 (pow (sin y) 2.0)) 2.0)
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
         3.0
         (* 1.1458980337503155 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = t_1 * sqrt(2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -0.95) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_1)) / (3.0 * (1.0 + fma(0.5, (cos(y) * t_3), (0.5 * t_0))));
	} else if (y <= 0.047) {
		tmp = (fma(t_2, (fma(sin(x), -0.0625, y) * fma(y, -0.0625, sin(x))), 2.0) * 0.3333333333333333) / ((fma(t_0, cos(x), (t_3 * cos(y))) / 2.0) + 1.0);
	} else {
		tmp = fma(t_2, (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(t_1 * sqrt(2.0))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -0.95)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_1)) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(y) * t_3), Float64(0.5 * t_0)))));
	elseif (y <= 0.047)
		tmp = Float64(Float64(fma(t_2, Float64(fma(sin(x), -0.0625, y) * fma(y, -0.0625, sin(x))), 2.0) * 0.3333333333333333) / Float64(Float64(fma(t_0, cos(x), Float64(t_3 * cos(y))) / 2.0) + 1.0));
	else
		tmp = Float64(fma(t_2, Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.95], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$3), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.047], N[(N[(N[(t$95$2 * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + y), $MachinePrecision] * N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.94999999999999996

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 59.8%

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

   4. Applied rewrites 59.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-sin.f64 59.7%

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

   7. Applied rewrites 59.7%

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

   if -0.94999999999999996 < y < 0.047

   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 99.2%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 50.2%

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

   if 0.047 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin y}^{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-sin.f64 62.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   9. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.95:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}\\ \mathbf{elif}\;y \leq 0.047:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), t\_0, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y))))
   (if (<= y -0.95)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0))
      (*
       3.0
       (+
        1.0
        (fma 0.5 (* (cos y) (- 3.0 (sqrt 5.0))) (* 0.5 (- (sqrt 5.0) 1.0))))))
     (if (<= y 0.047)
       (/
        (*
         (fma
          (* (- y (* (sin x) 0.0625)) (* (- (sin x) (* y 0.0625)) (sqrt 2.0)))
          t_0
          2.0)
         0.3333333333333333)
        (fma
         0.38196601125010515
         (cos y)
         (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
       (/
        (fma (* t_0 (sqrt 2.0)) (* -0.0625 (pow (sin y) 2.0)) 2.0)
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
         3.0
         (* 1.1458980337503155 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double tmp;
	if (y <= -0.95) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / (3.0 * (1.0 + fma(0.5, (cos(y) * (3.0 - sqrt(5.0))), (0.5 * (sqrt(5.0) - 1.0)))));
	} else if (y <= 0.047) {
		tmp = (fma(((y - (sin(x) * 0.0625)) * ((sin(x) - (y * 0.0625)) * sqrt(2.0))), t_0, 2.0) * 0.3333333333333333) / fma(0.38196601125010515, cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0));
	} else {
		tmp = fma((t_0 * sqrt(2.0)), (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (y <= -0.95)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0)) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(y) * Float64(3.0 - sqrt(5.0))), Float64(0.5 * Float64(sqrt(5.0) - 1.0))))));
	elseif (y <= 0.047)
		tmp = Float64(Float64(fma(Float64(Float64(y - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(y * 0.0625)) * sqrt(2.0))), t_0, 2.0) * 0.3333333333333333) / fma(0.38196601125010515, cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)));
	else
		tmp = Float64(fma(Float64(t_0 * sqrt(2.0)), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.95], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.047], N[(N[(N[(N[(N[(y - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.38196601125010515 * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.94999999999999996

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 59.8%

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

   4. Applied rewrites 59.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-sin.f64 59.7%

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

   7. Applied rewrites 59.7%

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

   if -0.94999999999999996 < y < 0.047

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.2%

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

   10. Applied rewrites 50.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), \cos x - \cos y, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]

   if 0.047 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin y}^{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-sin.f64 62.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   9. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.049:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.047:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), t\_0, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, -0.0625 \cdot t\_1, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y))) (t_1 (pow (sin y) 2.0)))
   (if (<= y -0.049)
     (/
      (+ 2.0 (* (* -0.0625 (* t_1 (sqrt 2.0))) t_0))
      (*
       3.0
       (+
        (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
        (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
     (if (<= y 0.047)
       (/
        (*
         (fma
          (* (- y (* (sin x) 0.0625)) (* (- (sin x) (* y 0.0625)) (sqrt 2.0)))
          t_0
          2.0)
         0.3333333333333333)
        (fma
         0.38196601125010515
         (cos y)
         (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
       (/
        (fma (* t_0 (sqrt 2.0)) (* -0.0625 t_1) 2.0)
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
         3.0
         (* 1.1458980337503155 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.049) {
		tmp = (2.0 + ((-0.0625 * (t_1 * sqrt(2.0))) * t_0)) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	} else if (y <= 0.047) {
		tmp = (fma(((y - (sin(x) * 0.0625)) * ((sin(x) - (y * 0.0625)) * sqrt(2.0))), t_0, 2.0) * 0.3333333333333333) / fma(0.38196601125010515, cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0));
	} else {
		tmp = fma((t_0 * sqrt(2.0)), (-0.0625 * t_1), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.049)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_1 * sqrt(2.0))) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	elseif (y <= 0.047)
		tmp = Float64(Float64(fma(Float64(Float64(y - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(y * 0.0625)) * sqrt(2.0))), t_0, 2.0) * 0.3333333333333333) / fma(0.38196601125010515, cos(y), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)));
	else
		tmp = Float64(fma(Float64(t_0 * sqrt(2.0)), Float64(-0.0625 * t_1), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.049], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $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], If[LessEqual[y, 0.047], N[(N[(N[(N[(N[(y - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.38196601125010515 * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.049000000000000002

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3%

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

   4. Applied rewrites 62.3%

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

   if -0.049000000000000002 < y < 0.047

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.2%

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

   10. Applied rewrites 50.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), \cos x - \cos y, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]

   if 0.047 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin y}^{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-sin.f64 62.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   9. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_2 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.049:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.047:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - t\_1, \left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, t\_1, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, -0.0625 \cdot t\_2, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (fma (* y y) -0.5 1.0))
        (t_2 (pow (sin y) 2.0)))
   (if (<= y -0.049)
     (/
      (+ 2.0 (* (* -0.0625 (* t_2 (sqrt 2.0))) t_0))
      (*
       3.0
       (+
        (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
        (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
     (if (<= y 0.047)
       (/
        (*
         (fma
          (- (cos x) t_1)
          (* (- y (* (sin x) 0.0625)) (* (- (sin x) (* y 0.0625)) (sqrt 2.0)))
          2.0)
         0.3333333333333333)
        (fma
         0.38196601125010515
         t_1
         (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
       (/
        (fma (* t_0 (sqrt 2.0)) (* -0.0625 t_2) 2.0)
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
         3.0
         (* 1.1458980337503155 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma((y * y), -0.5, 1.0);
	double t_2 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.049) {
		tmp = (2.0 + ((-0.0625 * (t_2 * sqrt(2.0))) * t_0)) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	} else if (y <= 0.047) {
		tmp = (fma((cos(x) - t_1), ((y - (sin(x) * 0.0625)) * ((sin(x) - (y * 0.0625)) * sqrt(2.0))), 2.0) * 0.3333333333333333) / fma(0.38196601125010515, t_1, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0));
	} else {
		tmp = fma((t_0 * sqrt(2.0)), (-0.0625 * t_2), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(Float64(y * y), -0.5, 1.0)
	t_2 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.049)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_2 * sqrt(2.0))) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	elseif (y <= 0.047)
		tmp = Float64(Float64(fma(Float64(cos(x) - t_1), Float64(Float64(y - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(y * 0.0625)) * sqrt(2.0))), 2.0) * 0.3333333333333333) / fma(0.38196601125010515, t_1, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)));
	else
		tmp = Float64(fma(Float64(t_0 * sqrt(2.0)), Float64(-0.0625 * t_2), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.049], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $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], If[LessEqual[y, 0.047], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - t$95$1), $MachinePrecision] * N[(N[(y - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.38196601125010515 * t$95$1 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.049000000000000002

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3%

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

   4. Applied rewrites 62.3%

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

   if -0.049000000000000002 < y < 0.047

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.2%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 49.7%

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

   11. Applied rewrites 49.7%

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

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 50.6%

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

   14. Applied rewrites 50.6%

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

   16. Applied rewrites 50.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \mathsf{fma}\left(y \cdot y, -0.5, 1\right), \left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \mathsf{fma}\left(y \cdot y, -0.5, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]

   if 0.047 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin y}^{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-sin.f64 62.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   9. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_2 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.049:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.047:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - t\_1, \left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, t\_1, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, -0.0625 \cdot t\_2, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (fma (* y y) -0.5 1.0))
        (t_2 (pow (sin y) 2.0)))
   (if (<= y -0.049)
     (/
      (+ 2.0 (* (* -0.0625 (* t_2 (sqrt 2.0))) t_0))
      (*
       3.0
       (+
        (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
        (* 0.38196601125010515 (cos y)))))
     (if (<= y 0.047)
       (/
        (*
         (fma
          (- (cos x) t_1)
          (* (- y (* (sin x) 0.0625)) (* (- (sin x) (* y 0.0625)) (sqrt 2.0)))
          2.0)
         0.3333333333333333)
        (fma
         0.38196601125010515
         t_1
         (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
       (/
        (fma (* t_0 (sqrt 2.0)) (* -0.0625 t_2) 2.0)
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
         3.0
         (* 1.1458980337503155 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma((y * y), -0.5, 1.0);
	double t_2 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.049) {
		tmp = (2.0 + ((-0.0625 * (t_2 * sqrt(2.0))) * t_0)) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (0.38196601125010515 * cos(y))));
	} else if (y <= 0.047) {
		tmp = (fma((cos(x) - t_1), ((y - (sin(x) * 0.0625)) * ((sin(x) - (y * 0.0625)) * sqrt(2.0))), 2.0) * 0.3333333333333333) / fma(0.38196601125010515, t_1, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0));
	} else {
		tmp = fma((t_0 * sqrt(2.0)), (-0.0625 * t_2), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(Float64(y * y), -0.5, 1.0)
	t_2 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.049)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_2 * sqrt(2.0))) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(0.38196601125010515 * cos(y)))));
	elseif (y <= 0.047)
		tmp = Float64(Float64(fma(Float64(cos(x) - t_1), Float64(Float64(y - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(y * 0.0625)) * sqrt(2.0))), 2.0) * 0.3333333333333333) / fma(0.38196601125010515, t_1, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)));
	else
		tmp = Float64(fma(Float64(t_0 * sqrt(2.0)), Float64(-0.0625 * t_2), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.049], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $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[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.047], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - t$95$1), $MachinePrecision] * N[(N[(y - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.38196601125010515 * t$95$1 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.049000000000000002

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.3%

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

   5. Applied rewrites 62.3%

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

   if -0.049000000000000002 < y < 0.047

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.2%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 49.7%

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

   11. Applied rewrites 49.7%

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

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 50.6%

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

   14. Applied rewrites 50.6%

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

   16. Applied rewrites 50.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \mathsf{fma}\left(y \cdot y, -0.5, 1\right), \left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \mathsf{fma}\left(y \cdot y, -0.5, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]

   if 0.047 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin y}^{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-sin.f64 62.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   9. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 79.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_1 := \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.049:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.047:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - t\_0, \left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* y y) -0.5 1.0))
        (t_1
         (/
          (fma
           (* (- (cos x) (cos y)) (sqrt 2.0))
           (* -0.0625 (pow (sin y) 2.0))
           2.0)
          (fma
           (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
           3.0
           (* 1.1458980337503155 (cos y))))))
   (if (<= y -0.049)
     t_1
     (if (<= y 0.047)
       (/
        (*
         (fma
          (- (cos x) t_0)
          (* (- y (* (sin x) 0.0625)) (* (- (sin x) (* y 0.0625)) (sqrt 2.0)))
          2.0)
         0.3333333333333333)
        (fma
         0.38196601125010515
         t_0
         (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), -0.5, 1.0);
	double t_1 = fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	double tmp;
	if (y <= -0.049) {
		tmp = t_1;
	} else if (y <= 0.047) {
		tmp = (fma((cos(x) - t_0), ((y - (sin(x) * 0.0625)) * ((sin(x) - (y * 0.0625)) * sqrt(2.0))), 2.0) * 0.3333333333333333) / fma(0.38196601125010515, t_0, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), -0.5, 1.0)
	t_1 = Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))))
	tmp = 0.0
	if (y <= -0.049)
		tmp = t_1;
	elseif (y <= 0.047)
		tmp = Float64(Float64(fma(Float64(cos(x) - t_0), Float64(Float64(y - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(y * 0.0625)) * sqrt(2.0))), 2.0) * 0.3333333333333333) / fma(0.38196601125010515, t_0, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.049], t$95$1, If[LessEqual[y, 0.047], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - t$95$0), $MachinePrecision] * N[(N[(y - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.38196601125010515 * t$95$0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.049000000000000002 or 0.047 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \frac{-1}{16} \cdot {\sin y}^{\color{blue}{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-sin.f64 62.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   9. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot {\sin y}^{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   if -0.049000000000000002 < y < 0.047

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.2%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 49.7%

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

   11. Applied rewrites 49.7%

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

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 50.6%

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

   14. Applied rewrites 50.6%

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

   16. Applied rewrites 50.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \mathsf{fma}\left(y \cdot y, -0.5, 1\right), \left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \mathsf{fma}\left(y \cdot y, -0.5, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 79.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\\ t_1 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ \mathbf{if}\;y \leq -0.049:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.047:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - t\_1, \left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, t\_1, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y)))))))
        (t_1 (fma (* y y) -0.5 1.0)))
   (if (<= y -0.049)
     (/
      t_0
      (*
       3.0
       (+
        (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
        (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
     (if (<= y 0.047)
       (/
        (*
         (fma
          (- (cos x) t_1)
          (* (- y (* (sin x) 0.0625)) (* (- (sin x) (* y 0.0625)) (sqrt 2.0)))
          2.0)
         0.3333333333333333)
        (fma
         0.38196601125010515
         t_1
         (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
       (/
        t_0
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
         3.0
         (* 1.1458980337503155 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))));
	double t_1 = fma((y * y), -0.5, 1.0);
	double tmp;
	if (y <= -0.049) {
		tmp = t_0 / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	} else if (y <= 0.047) {
		tmp = (fma((cos(x) - t_1), ((y - (sin(x) * 0.0625)) * ((sin(x) - (y * 0.0625)) * sqrt(2.0))), 2.0) * 0.3333333333333333) / fma(0.38196601125010515, t_1, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0));
	} else {
		tmp = t_0 / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))))
	t_1 = fma(Float64(y * y), -0.5, 1.0)
	tmp = 0.0
	if (y <= -0.049)
		tmp = Float64(t_0 / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	elseif (y <= 0.047)
		tmp = Float64(Float64(fma(Float64(cos(x) - t_1), Float64(Float64(y - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(y * 0.0625)) * sqrt(2.0))), 2.0) * 0.3333333333333333) / fma(0.38196601125010515, t_1, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)));
	else
		tmp = Float64(t_0 / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[y, -0.049], N[(t$95$0 / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.047], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - t$95$1), $MachinePrecision] * N[(N[(y - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.38196601125010515 * t$95$1 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.049000000000000002

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 62.2%

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

   4. Applied rewrites 62.2%

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

   if -0.049000000000000002 < y < 0.047

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.2%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 49.7%

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

   11. Applied rewrites 49.7%

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

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 50.6%

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

   14. Applied rewrites 50.6%

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

   16. Applied rewrites 50.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \mathsf{fma}\left(y \cdot y, -0.5, 1\right), \left(y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \mathsf{fma}\left(y \cdot y, -0.5, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]

   if 0.047 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      9. lower-cos.f64 62.3%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Applied rewrites 62.3%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\\ \mathbf{if}\;x \leq -43:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (*
           0.3333333333333333
           (+
            2.0
            (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
          (+
           (/
            (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
            2.0)
           1.0))))
   (if (<= x -43.0)
     t_0
     (if (<= x 1.6e-10)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.1458980337503155 (cos y) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0))))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (0.3333333333333333 * (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0)))))) / ((fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))) / 2.0) + 1.0);
	double tmp;
	if (x <= -43.0) {
		tmp = t_0;
	} else if (x <= 1.6e-10) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))) / Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) / 2.0) + 1.0))
	tmp = 0.0
	if (x <= -43.0)
		tmp = t_0;
	elseif (x <= 1.6e-10)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.3333333333333333 * 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]), $MachinePrecision] / N[(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] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -43.0], t$95$0, If[LessEqual[x, 1.6e-10], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.1458980337503155 * N[Cos[y], $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -43 or 1.5999999999999999e-10 < 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 99.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 62.2%

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

   6. Applied rewrites 62.2%

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

   if -43 < x < 1.5999999999999999e-10

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   7. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -43:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
          (fma
           (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
           3.0
           (* 1.1458980337503155 (cos y))))))
   (if (<= x -43.0)
     t_0
     (if (<= x 1.6e-10)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.1458980337503155 (cos y) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0))))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	double tmp;
	if (x <= -43.0) {
		tmp = t_0;
	} else if (x <= 1.6e-10) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))))
	tmp = 0.0
	if (x <= -43.0)
		tmp = t_0;
	elseif (x <= 1.6e-10)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -43.0], t$95$0, If[LessEqual[x, 1.6e-10], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.1458980337503155 * N[Cos[y], $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -43 or 1.5999999999999999e-10 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right)\right)} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      9. lower-cos.f64 62.2%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   9. Applied rewrites 62.2%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   if -43 < x < 1.5999999999999999e-10

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   7. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -15500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 65:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
          (fma
           (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
           3.0
           (* 1.1458980337503155 (cos y))))))
   (if (<= y -15500.0)
     t_0
     (if (<= y 65.0)
       (/
        (*
         0.3333333333333333
         (fma
          (* (- (cos x) 1.0) (sqrt 2.0))
          (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
          2.0))
        (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 1.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.1458980337503155 * cos(y)));
	double tmp;
	if (y <= -15500.0) {
		tmp = t_0;
	} else if (y <= 65.0) {
		tmp = (0.3333333333333333 * fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0)) / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.1458980337503155 * cos(y))))
	tmp = 0.0
	if (y <= -15500.0)
		tmp = t_0;
	elseif (y <= 65.0)
		tmp = Float64(Float64(0.3333333333333333 * fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0)) / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.1458980337503155 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -15500.0], t$95$0, If[LessEqual[y, 65.0], N[(N[(0.3333333333333333 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]), $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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -15500 or 65 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{20642663831210019}{18014398509481984} \cdot \cos y\right)} \]

      9. lower-cos.f64 62.3%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   7. Applied rewrites 62.3%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)} \]

   if -15500 < y < 65

   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 59.9%

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

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound--.f64 N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-unsound--.f64 N/A

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

      11. metadata-eval 59.8%

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

   6. Applied rewrites 59.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 59.9%

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

Alternative 22: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right)\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(t\_1, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(0.5, t\_0, 1\right)}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.0625 \cdot t\_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right) \cdot \left(\frac{1}{\mathsf{fma}\left(t\_0, 0.5, 1\right)} \cdot 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -1.5)
     (/
      (*
       0.3333333333333333
       (fma t_1 (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625) 2.0))
      (fma 0.5 t_0 1.0))
     (if (<= x 1.6e-10)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.1458980337503155 (cos y) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0))))))
       (*
        (fma (* -0.0625 t_1) (- 0.5 (* 0.5 (cos (* 2.0 x)))) 2.0)
        (* (/ 1.0 (fma t_0 0.5 1.0)) 0.3333333333333333))))))

C

double code(double x, double y) {
	double t_0 = fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0)));
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -1.5) {
		tmp = (0.3333333333333333 * fma(t_1, ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0)) / fma(0.5, t_0, 1.0);
	} else if (x <= 1.6e-10) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	} else {
		tmp = fma((-0.0625 * t_1), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) * ((1.0 / fma(t_0, 0.5, 1.0)) * 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0)))
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -1.5)
		tmp = Float64(Float64(0.3333333333333333 * fma(t_1, Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0)) / fma(0.5, t_0, 1.0));
	elseif (x <= 1.6e-10)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	else
		tmp = Float64(fma(Float64(-0.0625 * t_1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0) * Float64(Float64(1.0 / fma(t_0, 0.5, 1.0)) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5], N[(N[(0.3333333333333333 * N[(t$95$1 * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-10], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.1458980337503155 * N[Cos[y], $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(1.0 / N[(t$95$0 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   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 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 59.9%

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

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound--.f64 N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-unsound--.f64 N/A

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

      11. metadata-eval 59.8%

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

   6. Applied rewrites 59.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 59.9%

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

   if -1.5 < x < 1.5999999999999999e-10

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   7. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}} \]

   if 1.5999999999999999e-10 < 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 59.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 59.8%

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

Alternative 23: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.3333333333333333 \cdot \mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (*
           0.3333333333333333
           (fma
            (* (- (cos x) 1.0) (sqrt 2.0))
            (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
            2.0))
          (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 1.0))))
   (if (<= x -1.5)
     t_0
     (if (<= x 1.6e-10)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.1458980337503155 (cos y) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0))))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (0.3333333333333333 * fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0)) / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 1.0);
	double tmp;
	if (x <= -1.5) {
		tmp = t_0;
	} else if (x <= 1.6e-10) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(0.3333333333333333 * fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0)) / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 1.0))
	tmp = 0.0
	if (x <= -1.5)
		tmp = t_0;
	elseif (x <= 1.6e-10)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.3333333333333333 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5], t$95$0, If[LessEqual[x, 1.6e-10], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.1458980337503155 * N[Cos[y], $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 1.5999999999999999e-10 < 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 59.9%

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

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound--.f64 N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-unsound--.f64 N/A

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

      11. metadata-eval 59.8%

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

   6. Applied rewrites 59.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 59.9%

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

   if -1.5 < x < 1.5999999999999999e-10

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   7. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := 0.3333333333333333 \cdot \mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)\\ t_1 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{t\_0}{t\_1}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{t\_1}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          0.3333333333333333
          (fma
           (* (- (cos x) 1.0) (sqrt 2.0))
           (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
           2.0)))
        (t_1
         (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 1.0)))
   (if (<= x -1.5)
     (/ t_0 t_1)
     (if (<= x 1.6e-10)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.1458980337503155 (cos y) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0))))))
       (* t_0 (/ 1.0 t_1))))))

C

double code(double x, double y) {
	double t_0 = 0.3333333333333333 * fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0);
	double t_1 = fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 1.0);
	double tmp;
	if (x <= -1.5) {
		tmp = t_0 / t_1;
	} else if (x <= 1.6e-10) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	} else {
		tmp = t_0 * (1.0 / t_1);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.3333333333333333 * fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0))
	t_1 = fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 1.0)
	tmp = 0.0
	if (x <= -1.5)
		tmp = Float64(t_0 / t_1);
	elseif (x <= 1.6e-10)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.1458980337503155, cos(y), Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	else
		tmp = Float64(t_0 * Float64(1.0 / t_1));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.5], N[(t$95$0 / t$95$1), $MachinePrecision], If[LessEqual[x, 1.6e-10], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.1458980337503155 * N[Cos[y], $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   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 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 59.9%

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

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound--.f64 N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-unsound--.f64 N/A

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

      11. metadata-eval 59.8%

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

   6. Applied rewrites 59.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 59.9%

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

   if -1.5 < x < 1.5999999999999999e-10

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.1458980337503155 \cdot \cos y\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{20642663831210019}{18014398509481984} \cdot \cos y + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   7. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.1458980337503155, \cos y, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}} \]

   if 1.5999999999999999e-10 < 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 59.9%

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

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

      3. flip-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound--.f64 N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-unsound--.f64 N/A

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

      11. metadata-eval 59.8%

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

   6. Applied rewrites 59.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 59.8%

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

Alternative 25: 59.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(Float64(0.3333333333333333 * fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0)) / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 1.0))
end

Wolfram

code[x_, y_] := N[(N[(0.3333333333333333 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 59.9%

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

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

   3. flip-+ N/A

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

   4. lower-unsound-/.f64 N/A

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

   5. lower-unsound--.f64 N/A

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

   6. lower-unsound-*.f64 N/A

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

   7. lower-unsound-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-unsound--.f64 N/A

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

   11. metadata-eval 59.8%

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

6. Applied rewrites 59.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

8. Applied rewrites 59.8%

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

10. Applied rewrites 59.9%

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

Alternative 26: 59.9% accurate, 2.2× speedup

Math

\[0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot 1.2360679774997898, 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (fma
    (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
    (* (- (cos x) 1.0) (sqrt 2.0))
    2.0)
   (+
    1.0
    (fma 0.5 (* (cos x) 1.2360679774997898) (* 0.5 (- 3.0 (sqrt 5.0))))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / (1.0 + fma(0.5, (cos(x) * 1.2360679774997898), (0.5 * (3.0 - sqrt(5.0))))));
}

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / Float64(1.0 + fma(0.5, Float64(cos(x) * 1.2360679774997898), Float64(0.5 * Float64(3.0 - sqrt(5.0)))))))
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * 1.2360679774997898), $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 59.9%

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

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

   3. flip-+ N/A

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

   4. lower-unsound-/.f64 N/A

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

   5. lower-unsound--.f64 N/A

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

   6. lower-unsound-*.f64 N/A

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

   7. lower-unsound-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-unsound--.f64 N/A

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

   11. metadata-eval 59.8%

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

6. Applied rewrites 59.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

8. Applied rewrites 59.8%

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

9. Evaluated real constant 59.9%

   \[\leadsto 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot 1.2360679774997898, 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
10. Add Preprocessing

Alternative 27: 42.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(2.0 / Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(sqrt(5.0) - 1.0)), Float64(0.5 * Float64(3.0 - sqrt(5.0)))))))
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(2.0 / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 59.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 42.7%

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

Alternative 28: 40.2% accurate, 316.7× speedup

Math

\[0.3333333333333333 \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 59.9%

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

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. Evaluated real constant 40.2%

   \[\leadsto 0.3333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025184 
(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))))))