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

Percentage Accurate: 99.3% → 99.3%

Time: 14.4s

Alternatives: 25

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in y around inf

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

9. Applied rewrites 99.3%

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

10. Taylor expanded in x around inf

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

12. Applied rewrites 99.3%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $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 x around inf

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

4. Applied rewrites 99.2%

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

6. Applied rewrites 99.2%

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

Alternative 3: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\\ t_1 := \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.00175:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
           (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
        (t_1
         (/
          (+
           2.0
           (*
            (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
            (- (cos x) (cos y))))
          t_0)))
   (if (<= x -0.00175)
     t_1
     (if (<= x 2.5e-38)
       (/
        (+
         2.0
         (*
          (sqrt 2.0)
          (*
           (- 1.0 (cos y))
           (* (- (sin x) (* 0.0625 (sin y))) (- (sin y) (* 0.0625 (sin x)))))))
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y)));
	double t_1 = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / t_0;
	double tmp;
	if (x <= -0.00175) {
		tmp = t_1;
	} else if (x <= 2.5e-38) {
		tmp = (2.0 + (sqrt(2.0) * ((1.0 - cos(y)) * ((sin(x) - (0.0625 * sin(y))) * (sin(y) - (0.0625 * sin(x))))))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 * ((1.0d0 + (((sqrt(5.0d0) - 1.0d0) / 2.0d0) * cos(x))) + (((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y)))
    t_1 = (2.0d0 + (((sin(x) * sqrt(2.0d0)) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) - cos(y)))) / t_0
    if (x <= (-0.00175d0)) then
        tmp = t_1
    else if (x <= 2.5d-38) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((1.0d0 - cos(y)) * ((sin(x) - (0.0625d0 * sin(y))) * (sin(y) - (0.0625d0 * sin(x))))))) / t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (((Math.sqrt(5.0) - 1.0) / 2.0) * Math.cos(x))) + (((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y)));
	double t_1 = (2.0 + (((Math.sin(x) * Math.sqrt(2.0)) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) - Math.cos(y)))) / t_0;
	double tmp;
	if (x <= -0.00175) {
		tmp = t_1;
	} else if (x <= 2.5e-38) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * ((Math.sin(x) - (0.0625 * Math.sin(y))) * (Math.sin(y) - (0.0625 * Math.sin(x))))))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * ((1.0 + (((math.sqrt(5.0) - 1.0) / 2.0) * math.cos(x))) + (((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y)))
	t_1 = (2.0 + (((math.sin(x) * math.sqrt(2.0)) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) - math.cos(y)))) / t_0
	tmp = 0
	if x <= -0.00175:
		tmp = t_1
	elif x <= 2.5e-38:
		tmp = (2.0 + (math.sqrt(2.0) * ((1.0 - math.cos(y)) * ((math.sin(x) - (0.0625 * math.sin(y))) * (math.sin(y) - (0.0625 * math.sin(x))))))) / t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, 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))))
	t_1 = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / t_0)
	tmp = 0.0
	if (x <= -0.00175)
		tmp = t_1;
	elseif (x <= 2.5e-38)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * Float64(sin(y) - Float64(0.0625 * sin(x))))))) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, 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)));
	t_1 = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / t_0;
	tmp = 0.0;
	if (x <= -0.00175)
		tmp = t_1;
	elseif (x <= 2.5e-38)
		tmp = (2.0 + (sqrt(2.0) * ((1.0 - cos(y)) * ((sin(x) - (0.0625 * sin(y))) * (sin(y) - (0.0625 * sin(x))))))) / t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{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]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.00175], t$95$1, If[LessEqual[x, 2.5e-38], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00175000000000000004 or 2.50000000000000017e-38 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 64.6%

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

   if -0.00175000000000000004 < x < 2.50000000000000017e-38

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.0%

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

Alternative 4: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\\ t_2 := \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(t\_1 + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -0.00175:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)}{3 \cdot \left(t\_1 + 0.5 \cdot \left(\cos y \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))))
        (t_2
         (/
          (+
           2.0
           (*
            (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
            (- (cos x) (cos y))))
          (* 3.0 (+ t_1 (* (/ t_0 2.0) (cos y)))))))
   (if (<= x -0.00175)
     t_2
     (if (<= x 2.5e-38)
       (/
        (+
         2.0
         (*
          (sqrt 2.0)
          (*
           (fma (sin y) -0.0625 (sin x))
           (* (- 1.0 (cos y)) (fma (sin x) -0.0625 (sin y))))))
        (* 3.0 (+ t_1 (* 0.5 (* (cos y) t_0)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x));
	double t_2 = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * (t_1 + ((t_0 / 2.0) * cos(y))));
	double tmp;
	if (x <= -0.00175) {
		tmp = t_2;
	} else if (x <= 2.5e-38) {
		tmp = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * ((1.0 - cos(y)) * fma(sin(x), -0.0625, sin(y)))))) / (3.0 * (t_1 + (0.5 * (cos(y) * t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)))
	t_2 = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(t_1 + Float64(Float64(t_0 / 2.0) * cos(y)))))
	tmp = 0.0
	if (x <= -0.00175)
		tmp = t_2;
	elseif (x <= 2.5e-38)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(Float64(1.0 - cos(y)) * fma(sin(x), -0.0625, sin(y)))))) / Float64(3.0 * Float64(t_1 + Float64(0.5 * Float64(cos(y) * t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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$2 = N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00175], t$95$2, If[LessEqual[x, 2.5e-38], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00175000000000000004 or 2.50000000000000017e-38 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 64.6%

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

   if -0.00175000000000000004 < x < 2.50000000000000017e-38

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 63.0%

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

Alternative 5: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := 1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\\ t_3 := \frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \left(t\_2 + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, x\right) \cdot \left(t\_1 \cdot \mathsf{fma}\left(x, -0.0625, \sin y\right)\right)\right)}{3 \cdot \left(t\_2 + 0.5 \cdot \left(\cos y \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))))
        (t_3
         (/
          (+
           2.0
           (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_1))
          (* 3.0 (+ t_2 (* (/ t_0 2.0) (cos y)))))))
   (if (<= x -0.03)
     t_3
     (if (<= x 2.5e-38)
       (/
        (+
         2.0
         (*
          (sqrt 2.0)
          (* (fma (sin y) -0.0625 x) (* t_1 (fma x -0.0625 (sin y))))))
        (* 3.0 (+ t_2 (* 0.5 (* (cos y) t_0)))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = 1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x));
	double t_3 = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_1)) / (3.0 * (t_2 + ((t_0 / 2.0) * cos(y))));
	double tmp;
	if (x <= -0.03) {
		tmp = t_3;
	} else if (x <= 2.5e-38) {
		tmp = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, x) * (t_1 * fma(x, -0.0625, sin(y)))))) / (3.0 * (t_2 + (0.5 * (cos(y) * t_0))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)))
	t_3 = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * Float64(t_2 + Float64(Float64(t_0 / 2.0) * cos(y)))))
	tmp = 0.0
	if (x <= -0.03)
		tmp = t_3;
	elseif (x <= 2.5e-38)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, x) * Float64(t_1 * fma(x, -0.0625, sin(y)))))) / Float64(3.0 * Float64(t_2 + Float64(0.5 * Float64(cos(y) * t_0)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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$3 = N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.03], t$95$3, If[LessEqual[x, 2.5e-38], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + x), $MachinePrecision] * N[(t$95$1 * N[(x * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.029999999999999999 or 2.50000000000000017e-38 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 64.6%

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

   if -0.029999999999999999 < x < 2.50000000000000017e-38

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 51.6%

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

   13. Taylor expanded in x around 0

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

   15. Applied rewrites 51.7%

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

Alternative 6: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ t_3 := 1 + \frac{t\_2}{2} \cdot \cos x\\ t_4 := 0.5 \cdot \left(\cos y \cdot t\_0\right)\\ \mathbf{if}\;y \leq -0.047:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t\_1 \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right)\right)}{3 \cdot \left(t\_3 + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 1.06:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(t\_1 \cdot \left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, t\_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(t\_1 \cdot \sin y\right)\right)}{3 \cdot \left(t\_3 + t\_4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (+ 1.0 (* (/ t_2 2.0) (cos x))))
        (t_4 (* 0.5 (* (cos y) t_0))))
   (if (<= y -0.047)
     (/
      (+ 2.0 (* (sqrt 2.0) (* t_1 (* (- (sin x) (* 0.0625 (sin y))) (sin y)))))
      (* 3.0 (+ t_3 (* (/ t_0 2.0) (cos y)))))
     (if (<= y 1.06)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (*
           (sqrt 2.0)
           (*
            t_1
            (*
             (fma y -0.0625 (sin x))
             (fma
              (sin x)
              -0.0625
              (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0)))))))))
         (+ 1.0 (fma 0.5 (* (cos x) t_2) t_4))))
       (/
        (+
         2.0
         (* (sqrt 2.0) (* (fma (sin y) -0.0625 (sin x)) (* t_1 (sin y)))))
        (* 3.0 (+ t_3 t_4)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 1.0 + ((t_2 / 2.0) * cos(x));
	double t_4 = 0.5 * (cos(y) * t_0);
	double tmp;
	if (y <= -0.047) {
		tmp = (2.0 + (sqrt(2.0) * (t_1 * ((sin(x) - (0.0625 * sin(y))) * sin(y))))) / (3.0 * (t_3 + ((t_0 / 2.0) * cos(y))));
	} else if (y <= 1.06) {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (t_1 * (fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, (y * (1.0 + (-0.16666666666666666 * pow(y, 2.0))))))))) / (1.0 + fma(0.5, (cos(x) * t_2), t_4)));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * (t_1 * sin(y))))) / (3.0 * (t_3 + t_4));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x)))
	t_4 = Float64(0.5 * Float64(cos(y) * t_0))
	tmp = 0.0
	if (y <= -0.047)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_1 * Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y))))) / Float64(3.0 * Float64(t_3 + Float64(Float64(t_0 / 2.0) * cos(y)))));
	elseif (y <= 1.06)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_1 * Float64(fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0))))))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), t_4))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(t_1 * sin(y))))) / Float64(3.0 * Float64(t_3 + t_4)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.047], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * N[(N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if 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. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.4%

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

   if -0.047 < y < 1.0600000000000001

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.1%

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

   9. Applied rewrites 52.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 51.9%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if 1.0600000000000001 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 64.4%

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

Alternative 7: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\\ t_3 := \frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(t\_0 \cdot \sin y\right)\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + t\_2\right)}\\ \mathbf{if}\;y \leq -0.047:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.06:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(t\_0 \cdot \left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0)))))
        (t_3
         (/
          (+
           2.0
           (* (sqrt 2.0) (* (fma (sin y) -0.0625 (sin x)) (* t_0 (sin y)))))
          (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) t_2)))))
   (if (<= y -0.047)
     t_3
     (if (<= y 1.06)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (*
           (sqrt 2.0)
           (*
            t_0
            (*
             (fma y -0.0625 (sin x))
             (fma
              (sin x)
              -0.0625
              (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0)))))))))
         (+ 1.0 (fma 0.5 (* (cos x) t_1) t_2))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.5 * (cos(y) * (3.0 - sqrt(5.0)));
	double t_3 = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * (t_0 * sin(y))))) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + t_2));
	double tmp;
	if (y <= -0.047) {
		tmp = t_3;
	} else if (y <= 1.06) {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (t_0 * (fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, (y * (1.0 + (-0.16666666666666666 * pow(y, 2.0))))))))) / (1.0 + fma(0.5, (cos(x) * t_1), t_2)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))
	t_3 = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(t_0 * sin(y))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + t_2)))
	tmp = 0.0
	if (y <= -0.047)
		tmp = t_3;
	elseif (y <= 1.06)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_0 * Float64(fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0))))))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), t_2))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.047], t$95$3, If[LessEqual[y, 1.06], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.047 or 1.0600000000000001 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 64.4%

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

   if -0.047 < y < 1.0600000000000001

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.1%

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

   9. Applied rewrites 52.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 51.9%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\\ t_1 := \cos x - \cos y\\ t_2 := 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(t\_1 \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right)\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.052:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.06:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(t\_1 \cdot \left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          1.0
          (fma
           0.5
           (* (cos x) (- (sqrt 5.0) 1.0))
           (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0)))))))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (*
          0.3333333333333333
          (/
           (+
            2.0
            (* (sqrt 2.0) (* t_1 (* (- (sin x) (* 0.0625 (sin y))) (sin y)))))
           t_0))))
   (if (<= y -0.052)
     t_2
     (if (<= y 1.06)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (*
           (sqrt 2.0)
           (*
            t_1
            (*
             (fma y -0.0625 (sin x))
             (fma
              (sin x)
              -0.0625
              (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0)))))))))
         t_0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 1.0 + fma(0.5, (cos(x) * (sqrt(5.0) - 1.0)), (0.5 * (cos(y) * (3.0 - sqrt(5.0)))));
	double t_1 = cos(x) - cos(y);
	double t_2 = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (t_1 * ((sin(x) - (0.0625 * sin(y))) * sin(y))))) / t_0);
	double tmp;
	if (y <= -0.052) {
		tmp = t_2;
	} else if (y <= 1.06) {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (t_1 * (fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, (y * (1.0 + (-0.16666666666666666 * pow(y, 2.0))))))))) / t_0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(sqrt(5.0) - 1.0)), Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_1 * Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y))))) / t_0))
	tmp = 0.0
	if (y <= -0.052)
		tmp = t_2;
	elseif (y <= 1.06)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_1 * Float64(fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0))))))))) / t_0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.052], t$95$2, If[LessEqual[y, 1.06], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * N[(N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0519999999999999976 or 1.0600000000000001 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.3%

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

   if -0.0519999999999999976 < y < 1.0600000000000001

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.1%

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

   9. Applied rewrites 52.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 51.9%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\\ t_3 := 3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + t\_2\right)\\ \mathbf{if}\;y \leq -0.108:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot t\_0\right)\right)}{t\_3}\\ \mathbf{elif}\;y \leq 1.06:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin y}^{2} \cdot t\_0\right)\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0)))))
        (t_3 (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) t_2))))
   (if (<= y -0.108)
     (/
      (+ 2.0 (* (sqrt 2.0) (* (fma (sin y) -0.0625 (sin x)) (* (sin y) t_0))))
      t_3)
     (if (<= y 1.06)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (*
           (sqrt 2.0)
           (*
            (- (cos x) (cos y))
            (*
             (fma y -0.0625 (sin x))
             (fma
              (sin x)
              -0.0625
              (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0)))))))))
         (+ 1.0 (fma 0.5 (* (cos x) t_1) t_2))))
       (/ (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (pow (sin y) 2.0) t_0)))) t_3)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.5 * (cos(y) * (3.0 - sqrt(5.0)));
	double t_3 = 3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + t_2);
	double tmp;
	if (y <= -0.108) {
		tmp = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * (sin(y) * t_0)))) / t_3;
	} else if (y <= 1.06) {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, (y * (1.0 + (-0.16666666666666666 * pow(y, 2.0))))))))) / (1.0 + fma(0.5, (cos(x) * t_1), t_2)));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(y), 2.0) * t_0)))) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))
	t_3 = Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + t_2))
	tmp = 0.0
	if (y <= -0.108)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * t_0)))) / t_3);
	elseif (y <= 1.06)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0))))))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), t_2))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(y) ^ 2.0) * t_0)))) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.108], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 1.06], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.107999999999999999

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.8%

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

   if -0.107999999999999999 < y < 1.0600000000000001

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.1%

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

   9. Applied rewrites 52.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 51.9%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if 1.0600000000000001 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.7%

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

Alternative 10: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\\ t_3 := 3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + t\_2\right)\\ \mathbf{if}\;y \leq -0.102:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot t\_0\right)\right)}{t\_3}\\ \mathbf{elif}\;y \leq 1.06:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\left(\cos x + 0.5 \cdot {y}^{2}\right) - 1\right) \cdot \left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin y}^{2} \cdot t\_0\right)\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0)))))
        (t_3 (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) t_2))))
   (if (<= y -0.102)
     (/
      (+ 2.0 (* (sqrt 2.0) (* (fma (sin y) -0.0625 (sin x)) (* (sin y) t_0))))
      t_3)
     (if (<= y 1.06)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (*
           (sqrt 2.0)
           (*
            (- (+ (cos x) (* 0.5 (pow y 2.0))) 1.0)
            (* (fma y -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y))))))
         (+ 1.0 (fma 0.5 (* (cos x) t_1) t_2))))
       (/ (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (pow (sin y) 2.0) t_0)))) t_3)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.5 * (cos(y) * (3.0 - sqrt(5.0)));
	double t_3 = 3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + t_2);
	double tmp;
	if (y <= -0.102) {
		tmp = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * (sin(y) * t_0)))) / t_3;
	} else if (y <= 1.06) {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (((cos(x) + (0.5 * pow(y, 2.0))) - 1.0) * (fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y)))))) / (1.0 + fma(0.5, (cos(x) * t_1), t_2)));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(y), 2.0) * t_0)))) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))
	t_3 = Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + t_2))
	tmp = 0.0
	if (y <= -0.102)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * t_0)))) / t_3);
	elseif (y <= 1.06)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(Float64(cos(x) + Float64(0.5 * (y ^ 2.0))) - 1.0) * Float64(fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y)))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), t_2))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(y) ^ 2.0) * t_0)))) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.102], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 1.06], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Cos[x], $MachinePrecision] + N[(0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.101999999999999993

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.8%

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

   if -0.101999999999999993 < y < 1.0600000000000001

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.1%

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

   9. Applied rewrites 52.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 51.6%

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

   if 1.0600000000000001 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.7%

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

Alternative 11: 80.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 1 + -0.5 \cdot {y}^{2}\\ t_2 := 1 - \cos y\\ t_3 := 3 - \sqrt{5}\\ t_4 := 3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + 0.5 \cdot \left(\cos y \cdot t\_3\right)\right)\\ \mathbf{if}\;y \leq -0.0085:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot t\_2\right)\right)}{t\_4}\\ \mathbf{elif}\;y \leq 1.06:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - t\_1\right) \cdot \left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_0, 0.5 \cdot \left(t\_1 \cdot t\_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin y}^{2} \cdot t\_2\right)\right)}{t\_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (+ 1.0 (* -0.5 (pow y 2.0))))
        (t_2 (- 1.0 (cos y)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4
         (* 3.0 (+ (+ 1.0 (* (/ t_0 2.0) (cos x))) (* 0.5 (* (cos y) t_3))))))
   (if (<= y -0.0085)
     (/
      (+ 2.0 (* (sqrt 2.0) (* (fma (sin y) -0.0625 (sin x)) (* (sin y) t_2))))
      t_4)
     (if (<= y 1.06)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (*
           (sqrt 2.0)
           (*
            (- (cos x) t_1)
            (* (fma y -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y))))))
         (+ 1.0 (fma 0.5 (* (cos x) t_0) (* 0.5 (* t_1 t_3))))))
       (/ (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (pow (sin y) 2.0) t_2)))) t_4)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 1.0 + (-0.5 * pow(y, 2.0));
	double t_2 = 1.0 - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = 3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + (0.5 * (cos(y) * t_3)));
	double tmp;
	if (y <= -0.0085) {
		tmp = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * (sin(y) * t_2)))) / t_4;
	} else if (y <= 1.06) {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * ((cos(x) - t_1) * (fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y)))))) / (1.0 + fma(0.5, (cos(x) * t_0), (0.5 * (t_1 * t_3)))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(y), 2.0) * t_2)))) / t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(1.0 + Float64(-0.5 * (y ^ 2.0)))
	t_2 = Float64(1.0 - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(0.5 * Float64(cos(y) * t_3))))
	tmp = 0.0
	if (y <= -0.0085)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * t_2)))) / t_4);
	elseif (y <= 1.06)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - t_1) * Float64(fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y)))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_0), Float64(0.5 * Float64(t_1 * t_3))))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(y) ^ 2.0) * t_2)))) / t_4);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0085], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[y, 1.06], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - t$95$1), $MachinePrecision] * N[(N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0085000000000000006

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.8%

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

   if -0.0085000000000000006 < y < 1.0600000000000001

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.1%

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

   9. Applied rewrites 52.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 51.6%

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

   13. Taylor expanded in y around 0

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

   15. Applied rewrites 51.3%

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

   if 1.0600000000000001 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.7%

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

Alternative 12: 80.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\\ t_3 := 3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + t\_2\right)\\ \mathbf{if}\;y \leq -0.038:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot t\_0\right)\right)}{t\_3}\\ \mathbf{elif}\;y \leq 1.06:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x + -0.0625 \cdot y\right) \cdot \left(y - 0.0625 \cdot \sin x\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin y}^{2} \cdot t\_0\right)\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0)))))
        (t_3 (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) t_2))))
   (if (<= y -0.038)
     (/
      (+ 2.0 (* (sqrt 2.0) (* (fma (sin y) -0.0625 (sin x)) (* (sin y) t_0))))
      t_3)
     (if (<= y 1.06)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (*
           (sqrt 2.0)
           (*
            (- (cos x) (cos y))
            (* (+ (sin x) (* -0.0625 y)) (- y (* 0.0625 (sin x)))))))
         (+ 1.0 (fma 0.5 (* (cos x) t_1) t_2))))
       (/ (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (pow (sin y) 2.0) t_0)))) t_3)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.5 * (cos(y) * (3.0 - sqrt(5.0)));
	double t_3 = 3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + t_2);
	double tmp;
	if (y <= -0.038) {
		tmp = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * (sin(y) * t_0)))) / t_3;
	} else if (y <= 1.06) {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * ((sin(x) + (-0.0625 * y)) * (y - (0.0625 * sin(x))))))) / (1.0 + fma(0.5, (cos(x) * t_1), t_2)));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(y), 2.0) * t_0)))) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))
	t_3 = Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + t_2))
	tmp = 0.0
	if (y <= -0.038)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * t_0)))) / t_3);
	elseif (y <= 1.06)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) + Float64(-0.0625 * y)) * Float64(y - Float64(0.0625 * sin(x))))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), t_2))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(y) ^ 2.0) * t_0)))) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.038], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 1.06], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * y), $MachinePrecision]), $MachinePrecision] * N[(y - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0379999999999999991

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.8%

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

   if -0.0379999999999999991 < y < 1.0600000000000001

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 51.2%

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

   if 1.0600000000000001 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.7%

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

Alternative 13: 80.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\\ t_3 := 3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + t\_2\right)\\ \mathbf{if}\;y \leq -0.0068:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot t\_0\right)\right)}{t\_3}\\ \mathbf{elif}\;y \leq 0.86:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \left(\left(\sin x + -0.0625 \cdot y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin y}^{2} \cdot t\_0\right)\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0)))))
        (t_3 (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) t_2))))
   (if (<= y -0.0068)
     (/
      (+ 2.0 (* (sqrt 2.0) (* (fma (sin y) -0.0625 (sin x)) (* (sin y) t_0))))
      t_3)
     (if (<= y 0.86)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (*
           (sqrt 2.0)
           (*
            (- (cos x) 1.0)
            (* (+ (sin x) (* -0.0625 y)) (- (sin y) (* 0.0625 (sin x)))))))
         (+ 1.0 (fma 0.5 (* (cos x) t_1) t_2))))
       (/ (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (pow (sin y) 2.0) t_0)))) t_3)))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.5 * (cos(y) * (3.0 - sqrt(5.0)));
	double t_3 = 3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + t_2);
	double tmp;
	if (y <= -0.0068) {
		tmp = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * (sin(y) * t_0)))) / t_3;
	} else if (y <= 0.86) {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * ((cos(x) - 1.0) * ((sin(x) + (-0.0625 * y)) * (sin(y) - (0.0625 * sin(x))))))) / (1.0 + fma(0.5, (cos(x) * t_1), t_2)));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(y), 2.0) * t_0)))) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))
	t_3 = Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + t_2))
	tmp = 0.0
	if (y <= -0.0068)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * t_0)))) / t_3);
	elseif (y <= 0.86)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - 1.0) * Float64(Float64(sin(x) + Float64(-0.0625 * y)) * Float64(sin(y) - Float64(0.0625 * sin(x))))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), t_2))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(y) ^ 2.0) * t_0)))) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0068], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 0.86], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * y), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00679999999999999962

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.8%

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

   if -0.00679999999999999962 < y < 0.859999999999999987

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 58.3%

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

   if 0.859999999999999987 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.7%

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

Alternative 14: 80.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 1 - \cos y\\ t_2 := 3 - \sqrt{5}\\ t_3 := 3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + 0.5 \cdot \left(\cos y \cdot t\_2\right)\right)\\ \mathbf{if}\;y \leq -0.0007:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot t\_1\right)\right)}{t\_3}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_0, 0.5 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin y}^{2} \cdot t\_1\right)\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 1.0 (cos y)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (* 3.0 (+ (+ 1.0 (* (/ t_0 2.0) (cos x))) (* 0.5 (* (cos y) t_2))))))
   (if (<= y -0.0007)
     (/
      (+ 2.0 (* (sqrt 2.0) (* (fma (sin y) -0.0625 (sin x)) (* (sin y) t_1))))
      t_3)
     (if (<= y 4.8e-5)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (*
           (sqrt 2.0)
           (*
            (- (cos x) (cos y))
            (* (fma y -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y))))))
         (+ 1.0 (fma 0.5 (* (cos x) t_0) (* 0.5 t_2)))))
       (/ (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (pow (sin y) 2.0) t_1)))) t_3)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 1.0 - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + (0.5 * (cos(y) * t_2)));
	double tmp;
	if (y <= -0.0007) {
		tmp = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * (sin(y) * t_1)))) / t_3;
	} else if (y <= 4.8e-5) {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y)))))) / (1.0 + fma(0.5, (cos(x) * t_0), (0.5 * t_2))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(y), 2.0) * t_1)))) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(1.0 - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(0.5 * Float64(cos(y) * t_2))))
	tmp = 0.0
	if (y <= -0.0007)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * t_1)))) / t_3);
	elseif (y <= 4.8e-5)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(fma(y, -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y)))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_0), Float64(0.5 * t_2)))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(y) ^ 2.0) * t_1)))) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0007], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 4.8e-5], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.99999999999999993e-4

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.8%

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

   if -6.99999999999999993e-4 < y < 4.8000000000000001e-5

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.1%

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

   9. Applied rewrites 52.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 51.8%

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

   if 4.8000000000000001e-5 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 62.7%

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

Alternative 15: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ t_2 := \sqrt{5} - 1\\ t_3 := 1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, 0.5 \cdot \left(\cos y \cdot t\_0\right)\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot t\_1\right)\right)}{t\_3}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.0625, {\sin y}^{2} \cdot \sqrt{2}, x \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot 1.00390625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot 1\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(t\_1 \cdot \left(\cos x - 1\right)\right)\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (+ 1.0 (fma 0.5 (* (cos x) t_2) (* 0.5 (* (cos y) t_0))))))
   (if (<= x -1.45e-5)
     (*
      0.3333333333333333
      (/ (+ 2.0 (* (sqrt 2.0) (* (- (cos x) (cos y)) (* -0.0625 t_1)))) t_3))
     (if (<= x 2.5e-38)
       (/
        (+
         2.0
         (*
          (fma
           -0.0625
           (* (pow (sin y) 2.0) (sqrt 2.0))
           (* x (* (sqrt 2.0) (* (sin y) 1.00390625))))
          (- 1.0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) 1.0)) (* (/ t_0 2.0) (cos y)))))
       (*
        0.3333333333333333
        (/ (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* t_1 (- (cos x) 1.0))))) t_3))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 1.0 + fma(0.5, (cos(x) * t_2), (0.5 * (cos(y) * t_0)));
	double tmp;
	if (x <= -1.45e-5) {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * t_1)))) / t_3);
	} else if (x <= 2.5e-38) {
		tmp = (2.0 + (fma(-0.0625, (pow(sin(y), 2.0) * sqrt(2.0)), (x * (sqrt(2.0) * (sin(y) * 1.00390625)))) * (1.0 - cos(y)))) / (3.0 * ((1.0 + ((t_2 / 2.0) * 1.0)) + ((t_0 / 2.0) * cos(y))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * (t_1 * (cos(x) - 1.0))))) / t_3);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), Float64(0.5 * Float64(cos(y) * t_0))))
	tmp = 0.0
	if (x <= -1.45e-5)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * t_1)))) / t_3));
	elseif (x <= 2.5e-38)
		tmp = Float64(Float64(2.0 + Float64(fma(-0.0625, Float64((sin(y) ^ 2.0) * sqrt(2.0)), Float64(x * Float64(sqrt(2.0) * Float64(sin(y) * 1.00390625)))) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * 1.0)) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64(t_1 * Float64(cos(x) - 1.0))))) / t_3));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e-5], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-38], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(t$95$1 * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-5

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 63.0%

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

   if -1.45e-5 < x < 2.50000000000000017e-38

   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

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 55.8%

       \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2} \cdot \sqrt{2}, x \cdot \left(\sqrt{2} \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)\right)} \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. Applied rewrites 55.8%

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

   if 2.50000000000000017e-38 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 63.0%

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

Alternative 16: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{t\_0}{2} \cdot \cos y\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{t\_2}{2}\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + t\_3 \cdot \cos x\right) + t\_1\right)}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.0625, {\sin y}^{2} \cdot \sqrt{2}, x \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot 1.00390625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_3 \cdot 1\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, 0.5 \cdot \left(\cos y \cdot t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (/ t_0 2.0) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (/ t_2 2.0)))
   (if (<= x -1.55e-5)
     (/
      (+
       2.0
       (*
        (* -0.0625 (* (- 0.5 (* 0.5 (cos (* 2.0 x)))) (sqrt 2.0)))
        (- (cos x) (cos y))))
      (* 3.0 (+ (+ 1.0 (* t_3 (cos x))) t_1)))
     (if (<= x 2.5e-38)
       (/
        (+
         2.0
         (*
          (fma
           -0.0625
           (* (pow (sin y) 2.0) (sqrt 2.0))
           (* x (* (sqrt 2.0) (* (sin y) 1.00390625))))
          (- 1.0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_3 1.0)) t_1)))
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* (sqrt 2.0) (* -0.0625 (* (pow (sin x) 2.0) (- (cos x) 1.0)))))
         (+ 1.0 (fma 0.5 (* (cos x) t_2) (* 0.5 (* (cos y) t_0))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (t_0 / 2.0) * cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = t_2 / 2.0;
	double tmp;
	if (x <= -1.55e-5) {
		tmp = (2.0 + ((-0.0625 * ((0.5 - (0.5 * cos((2.0 * x)))) * sqrt(2.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (t_3 * cos(x))) + t_1));
	} else if (x <= 2.5e-38) {
		tmp = (2.0 + (fma(-0.0625, (pow(sin(y), 2.0) * sqrt(2.0)), (x * (sqrt(2.0) * (sin(y) * 1.00390625)))) * (1.0 - cos(y)))) / (3.0 * ((1.0 + (t_3 * 1.0)) + t_1));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(x), 2.0) * (cos(x) - 1.0))))) / (1.0 + fma(0.5, (cos(x) * t_2), (0.5 * (cos(y) * t_0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(t_0 / 2.0) * cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(t_2 / 2.0)
	tmp = 0.0
	if (x <= -1.55e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * sqrt(2.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * cos(x))) + t_1)));
	elseif (x <= 2.5e-38)
		tmp = Float64(Float64(2.0 + Float64(fma(-0.0625, Float64((sin(y) ^ 2.0) * sqrt(2.0)), Float64(x * Float64(sqrt(2.0) * Float64(sin(y) * 1.00390625)))) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * 1.0)) + t_1)));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(cos(x) - 1.0))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), Float64(0.5 * Float64(cos(y) * t_0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.55e-5], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-38], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * 1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000007e-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

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

   4. Applied rewrites 63.0%

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

   6. Applied rewrites 63.0%

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

   if -1.55000000000000007e-5 < x < 2.50000000000000017e-38

   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

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 55.8%

       \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2} \cdot \sqrt{2}, x \cdot \left(\sqrt{2} \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)\right)} \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. Applied rewrites 55.8%

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

   if 2.50000000000000017e-38 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 63.0%

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

Alternative 17: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, 0.5 \cdot \left(\cos y \cdot t\_0\right)\right)}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.0625, {\sin y}^{2} \cdot \sqrt{2}, x \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot 1.00390625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot 1\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (*
          0.3333333333333333
          (/
           (+
            2.0
            (* (sqrt 2.0) (* -0.0625 (* (pow (sin x) 2.0) (- (cos x) 1.0)))))
           (+ 1.0 (fma 0.5 (* (cos x) t_1) (* 0.5 (* (cos y) t_0))))))))
   (if (<= x -1.45e-5)
     t_2
     (if (<= x 2.5e-38)
       (/
        (+
         2.0
         (*
          (fma
           -0.0625
           (* (pow (sin y) 2.0) (sqrt 2.0))
           (* x (* (sqrt 2.0) (* (sin y) 1.00390625))))
          (- 1.0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) 1.0)) (* (/ t_0 2.0) (cos y)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(x), 2.0) * (cos(x) - 1.0))))) / (1.0 + fma(0.5, (cos(x) * t_1), (0.5 * (cos(y) * t_0)))));
	double tmp;
	if (x <= -1.45e-5) {
		tmp = t_2;
	} else if (x <= 2.5e-38) {
		tmp = (2.0 + (fma(-0.0625, (pow(sin(y), 2.0) * sqrt(2.0)), (x * (sqrt(2.0) * (sin(y) * 1.00390625)))) * (1.0 - cos(y)))) / (3.0 * ((1.0 + ((t_1 / 2.0) * 1.0)) + ((t_0 / 2.0) * cos(y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(cos(x) - 1.0))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), Float64(0.5 * Float64(cos(y) * t_0))))))
	tmp = 0.0
	if (x <= -1.45e-5)
		tmp = t_2;
	elseif (x <= 2.5e-38)
		tmp = Float64(Float64(2.0 + Float64(fma(-0.0625, Float64((sin(y) ^ 2.0) * sqrt(2.0)), Float64(x * Float64(sqrt(2.0) * Float64(sin(y) * 1.00390625)))) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * 1.0)) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e-5], t$95$2, If[LessEqual[x, 2.5e-38], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e-5 or 2.50000000000000017e-38 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 63.0%

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

   if -1.45e-5 < x < 2.50000000000000017e-38

   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

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 55.8%

       \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2} \cdot \sqrt{2}, x \cdot \left(\sqrt{2} \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)\right)} \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. Applied rewrites 55.8%

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

Alternative 18: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, 0.5 \cdot \left(\cos y \cdot t\_0\right)\right)}\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.0625, {\sin y}^{2} \cdot \sqrt{2}, \left(2.0078125 \cdot \sin y\right) \cdot x\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot 1\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (*
          0.3333333333333333
          (/
           (+
            2.0
            (* (sqrt 2.0) (* -0.0625 (* (pow (sin x) 2.0) (- (cos x) 1.0)))))
           (+ 1.0 (fma 0.5 (* (cos x) t_1) (* 0.5 (* (cos y) t_0))))))))
   (if (<= x -1.85e-6)
     t_2
     (if (<= x 2.5e-38)
       (/
        (+
         2.0
         (*
          (fma
           -0.0625
           (* (pow (sin y) 2.0) (sqrt 2.0))
           (* (* 2.0078125 (sin y)) x))
          (- 1.0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) 1.0)) (* (/ t_0 2.0) (cos y)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(x), 2.0) * (cos(x) - 1.0))))) / (1.0 + fma(0.5, (cos(x) * t_1), (0.5 * (cos(y) * t_0)))));
	double tmp;
	if (x <= -1.85e-6) {
		tmp = t_2;
	} else if (x <= 2.5e-38) {
		tmp = (2.0 + (fma(-0.0625, (pow(sin(y), 2.0) * sqrt(2.0)), ((2.0078125 * sin(y)) * x)) * (1.0 - cos(y)))) / (3.0 * ((1.0 + ((t_1 / 2.0) * 1.0)) + ((t_0 / 2.0) * cos(y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(cos(x) - 1.0))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), Float64(0.5 * Float64(cos(y) * t_0))))))
	tmp = 0.0
	if (x <= -1.85e-6)
		tmp = t_2;
	elseif (x <= 2.5e-38)
		tmp = Float64(Float64(2.0 + Float64(fma(-0.0625, Float64((sin(y) ^ 2.0) * sqrt(2.0)), Float64(Float64(2.0078125 * sin(y)) * x)) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * 1.0)) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.85e-6], t$95$2, If[LessEqual[x, 2.5e-38], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0078125 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8500000000000001e-6 or 2.50000000000000017e-38 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 63.0%

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

   if -1.8500000000000001e-6 < x < 2.50000000000000017e-38

   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

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 55.8%

       \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2} \cdot \sqrt{2}, x \cdot \left(\sqrt{2} \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)\right)} \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. Applied rewrites 55.6%

       \[\leadsto \frac{2 + \mathsf{fma}\left(-0.0625, {\sin y}^{2} \cdot \sqrt{2}, \left(2.0078125 \cdot \sin y\right) \cdot x\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, 0.5 \cdot \left(\cos y \cdot t\_0\right)\right)}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) \cdot \sqrt{2}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot 1\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (*
          0.3333333333333333
          (/
           (+
            2.0
            (* (sqrt 2.0) (* -0.0625 (* (pow (sin x) 2.0) (- (cos x) 1.0)))))
           (+ 1.0 (fma 0.5 (* (cos x) t_1) (* 0.5 (* (cos y) t_0))))))))
   (if (<= x -1.7e-6)
     t_2
     (if (<= x 2.5e-38)
       (/
        (+
         2.0
         (*
          (* -0.0625 (* (- 0.5 (* 0.5 (cos (* 2.0 y)))) (sqrt 2.0)))
          (- 1.0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) 1.0)) (* (/ t_0 2.0) (cos y)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(x), 2.0) * (cos(x) - 1.0))))) / (1.0 + fma(0.5, (cos(x) * t_1), (0.5 * (cos(y) * t_0)))));
	double tmp;
	if (x <= -1.7e-6) {
		tmp = t_2;
	} else if (x <= 2.5e-38) {
		tmp = (2.0 + ((-0.0625 * ((0.5 - (0.5 * cos((2.0 * y)))) * sqrt(2.0))) * (1.0 - cos(y)))) / (3.0 * ((1.0 + ((t_1 / 2.0) * 1.0)) + ((t_0 / 2.0) * cos(y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(cos(x) - 1.0))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), Float64(0.5 * Float64(cos(y) * t_0))))))
	tmp = 0.0
	if (x <= -1.7e-6)
		tmp = t_2;
	elseif (x <= 2.5e-38)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y)))) * sqrt(2.0))) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * 1.0)) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e-6], t$95$2, If[LessEqual[x, 2.5e-38], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000003e-6 or 2.50000000000000017e-38 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 63.0%

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

   if -1.70000000000000003e-6 < x < 2.50000000000000017e-38

   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

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 59.8%

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

   12. Applied rewrites 59.8%

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

Alternative 20: 79.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_0, 0.5 \cdot t\_1\right)}\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) \cdot \sqrt{2}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot 1\right) + \frac{t\_1}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (*
          0.3333333333333333
          (/
           (+
            2.0
            (*
             -0.0625
             (*
              (- 0.5 (* 0.5 (cos (* 2.0 x))))
              (* (sqrt 2.0) (- (cos x) 1.0)))))
           (+ 1.0 (fma 0.5 (* (cos x) t_0) (* 0.5 t_1)))))))
   (if (<= x -1.85e-6)
     t_2
     (if (<= x 7.4e-6)
       (/
        (+
         2.0
         (*
          (* -0.0625 (* (- 0.5 (* 0.5 (cos (* 2.0 y)))) (sqrt 2.0)))
          (- 1.0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* (/ t_0 2.0) 1.0)) (* (/ t_1 2.0) (cos y)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 0.3333333333333333 * ((2.0 + (-0.0625 * ((0.5 - (0.5 * cos((2.0 * x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.0 + fma(0.5, (cos(x) * t_0), (0.5 * t_1))));
	double tmp;
	if (x <= -1.85e-6) {
		tmp = t_2;
	} else if (x <= 7.4e-6) {
		tmp = (2.0 + ((-0.0625 * ((0.5 - (0.5 * cos((2.0 * y)))) * sqrt(2.0))) * (1.0 - cos(y)))) / (3.0 * ((1.0 + ((t_0 / 2.0) * 1.0)) + ((t_1 / 2.0) * cos(y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_0), Float64(0.5 * t_1)))))
	tmp = 0.0
	if (x <= -1.85e-6)
		tmp = t_2;
	elseif (x <= 7.4e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y)))) * sqrt(2.0))) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * 1.0)) + Float64(Float64(t_1 / 2.0) * cos(y)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.85e-6], t$95$2, If[LessEqual[x, 7.4e-6], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8500000000000001e-6 or 7.4000000000000003e-6 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 60.8%

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

   if -1.8500000000000001e-6 < x < 7.4000000000000003e-6

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 59.8%

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

   12. Applied rewrites 59.8%

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

Alternative 21: 60.8% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+
    2.0
    (*
     -0.0625
     (* (- 0.5 (* 0.5 (cos (* 2.0 x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
   (+
    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 + (-0.0625 * ((0.5 - (0.5 * cos((2.0 * x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / (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(Float64(2.0 + Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / 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[(N[(2.0 + N[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(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

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

4. Applied rewrites 60.8%

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

6. Applied rewrites 60.8%

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

Alternative 22: 44.0% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+
    (fma
     (* -0.0625 (fma (cos x) 2.0 -2.0))
     (- 0.5 (* 0.5 (cos (* 2.0 x))))
     1.0)
    1.0)
   (+
    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 * ((fma((-0.0625 * fma(cos(x), 2.0, -2.0)), (0.5 - (0.5 * cos((2.0 * x)))), 1.0) + 1.0) / (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(Float64(fma(Float64(-0.0625 * fma(cos(x), 2.0, -2.0)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 1.0) + 1.0) / 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[(N[(N[(N[(-0.0625 * N[(N[Cos[x], $MachinePrecision] * 2.0 + -2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / 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

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

4. Applied rewrites 60.8%

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

6. Applied rewrites 44.0%

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

Alternative 23: 44.0% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (fma (fma (cos x) 2.0 -2.0) (* (- 0.5 (* 0.5 (cos (* 2.0 x)))) -0.0625) 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 * (fma(fma(cos(x), 2.0, -2.0), ((0.5 - (0.5 * cos((2.0 * x)))) * -0.0625), 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(fma(fma(cos(x), 2.0, -2.0), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * -0.0625), 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[(N[(N[(N[Cos[x], $MachinePrecision] * 2.0 + -2.0), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / 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

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

4. Applied rewrites 60.8%

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

6. Applied rewrites 44.0%

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

Alternative 24: 43.5% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(2.0 / Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(sqrt(5.0) - 1.0)), Float64(0.5 * Float64(3.0 - sqrt(5.0)))))))
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(2.0 / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 60.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 43.5%

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

Alternative 25: 41.0% accurate, 316.7× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 60.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 41.0%

   \[\leadsto \frac{0.6666666666666666}{\color{blue}{1 + \mathsf{fma}\left(0.5, 3 - \sqrt{5}, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]

9. Applied rewrites 41.0%

   \[\leadsto \frac{0.6666666666666666}{\color{blue}{\mathsf{fma}\left(-2 - -4, 0.5, 1\right)}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{1}{3} \]

12. Applied rewrites 41.0%

    \[\leadsto 0.3333333333333333 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025161 
(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))))))