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

Percentage Accurate: 99.3% → 99.4%

Time: 9.4s

Alternatives: 22

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\frac{\mathsf{fma}\left(\cos y - \cos x, \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), -1.5, -3\right)} \]

FPCore

(FPCore (x y)
  :precision binary64
  (/
 (fma
  (- (cos y) (cos x))
  (*
   (* (- (sin y) (* 0.0625 (sin x))) (sqrt 2.0))
   (- (sin x) (* 0.0625 (sin y))))
  -2.0)
 (fma
  (fma (- (sqrt 5.0) 1.0) (cos x) (* 0.7639320225002103 (cos y)))
  -1.5
  -3.0)))

C

double code(double x, double y) {
	return fma((cos(y) - cos(x)), (((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y)))), -2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), (0.7639320225002103 * cos(y))), -1.5, -3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(cos(y) - cos(x)), Float64(Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * sqrt(2.0)) * Float64(sin(x) - Float64(0.0625 * sin(y)))), -2.0) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(0.7639320225002103 * cos(y))), -1.5, -3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + -3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

3. Applied rewrites 99.3%

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

4. Evaluated real constant 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
5. Step-by-step derivation

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \left(\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), -1.5, -3\right)}} \]
9. Add Preprocessing

Alternative 2: 99.4% accurate, 1.1× speedup

Math

\[\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(1.2360679774997898, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

FPCore

(FPCore (x y)
  :precision binary64
  (/
 (fma
  (- (cos x) (cos y))
  (*
   (- (sin y) (* (sin x) 0.0625))
   (* (- (sin x) (* (sin y) 0.0625)) (sqrt 2.0)))
  2.0)
 (fma
  (*
   (fma 1.2360679774997898 (cos x) (* 0.7639320225002103 (cos y)))
   0.5)
  3.0
  3.0)))

C

double code(double x, double y) {
	return fma((cos(x) - cos(y)), ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (sin(y) * 0.0625)) * sqrt(2.0))), 2.0) / fma((fma(1.2360679774997898, cos(x), (0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * sqrt(2.0))), 2.0) / fma(Float64(fma(1.2360679774997898, cos(x), Float64(0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(1.2360679774997898 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

3. Applied rewrites 99.3%

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

4. Evaluated real constant 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
5. Step-by-step derivation

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

7. Evaluated real constant 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{1.2360679774997898}, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
8. Add Preprocessing

Alternative 3: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)\\ t_1 := \cos x - \cos y\\ t_2 := \sin y - \sin x \cdot 0.0625\\ t_3 := \frac{\mathsf{fma}\left(t\_1, t\_2 \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.043:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.76:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_2 \cdot \left(\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0
        (fma
         (*
          (fma
           (- (sqrt 5.0) 1.0)
           (cos x)
           (* 0.7639320225002103 (cos y)))
          0.5)
         3.0
         3.0))
       (t_1 (- (cos x) (cos y)))
       (t_2 (- (sin y) (* (sin x) 0.0625)))
       (t_3 (/ (fma t_1 (* t_2 (* (sin x) (sqrt 2.0))) 2.0) t_0)))
  (if (<= x -0.043)
    t_3
    (if (<= x 1.76)
      (/
       (fma
        t_1
        (*
         t_2
         (*
          (-
           (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0))))
           (* 0.0625 (sin y)))
          (sqrt 2.0)))
        2.0)
       t_0)
      t_3))))

C

double code(double x, double y) {
	double t_0 = fma((fma((sqrt(5.0) - 1.0), cos(x), (0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sin(y) - (sin(x) * 0.0625);
	double t_3 = fma(t_1, (t_2 * (sin(x) * sqrt(2.0))), 2.0) / t_0;
	double tmp;
	if (x <= -0.043) {
		tmp = t_3;
	} else if (x <= 1.76) {
		tmp = fma(t_1, (t_2 * (((x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)))) - (0.0625 * sin(y))) * sqrt(2.0))), 2.0) / t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sin(y) - Float64(sin(x) * 0.0625))
	t_3 = Float64(fma(t_1, Float64(t_2 * Float64(sin(x) * sqrt(2.0))), 2.0) / t_0)
	tmp = 0.0
	if (x <= -0.043)
		tmp = t_3;
	elseif (x <= 1.76)
		tmp = Float64(fma(t_1, Float64(t_2 * Float64(Float64(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0)))) - Float64(0.0625 * sin(y))) * sqrt(2.0))), 2.0) / t_0);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.043], t$95$3, If[LessEqual[x, 1.76], N[(N[(t$95$1 * N[(t$95$2 * N[(N[(N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.042999999999999997 or 1.76 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \color{blue}{\left(\sin x \cdot \sqrt{2}\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x \cdot \color{blue}{\sqrt{2}}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x \cdot \sqrt{\color{blue}{2}}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-sqrt.f64 63.9%

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 63.9%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \color{blue}{\left(\sin x \cdot \sqrt{2}\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -0.042999999999999997 < x < 1.76

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{1}{16} \cdot \sin y\right)} \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \color{blue}{\frac{1}{16} \cdot \sin y}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \color{blue}{\frac{1}{16}} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{1}{16} \cdot \color{blue}{\sin y}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      7. lower-sin.f64 51.0%

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 51.0%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right) - 0.0625 \cdot \sin y\right)} \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(t\_0, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}\\ \mathbf{if}\;x \leq -0.043:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, \left(\sin y - \mathsf{fma}\left(x \cdot x, -0.010416666666666666, 0.0625\right) \cdot x\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1.5, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) (cos y)))
       (t_1 (- (sqrt 5.0) 1.0))
       (t_2
        (/
         (fma
          t_0
          (* (- (sin y) (* (sin x) 0.0625)) (* (sin x) (sqrt 2.0)))
          2.0)
         (fma
          (* (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 0.5)
          3.0
          3.0))))
  (if (<= x -0.043)
    t_2
    (if (<= x 1.55)
      (/
       (fma
        (* t_0 (sqrt 2.0))
        (*
         (- (sin y) (* (fma (* x x) -0.010416666666666666 0.0625) x))
         (fma -0.0625 (sin y) (sin x)))
        2.0)
       (fma (fma t_1 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 1.5 3.0))
      t_2))))

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 = fma(t_0, ((sin(y) - (sin(x) * 0.0625)) * (sin(x) * sqrt(2.0))), 2.0) / fma((fma(t_1, cos(x), (0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0);
	double tmp;
	if (x <= -0.043) {
		tmp = t_2;
	} else if (x <= 1.55) {
		tmp = fma((t_0 * sqrt(2.0)), ((sin(y) - (fma((x * x), -0.010416666666666666, 0.0625) * x)) * fma(-0.0625, sin(y), sin(x))), 2.0) / fma(fma(t_1, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.5, 3.0);
	} else {
		tmp = t_2;
	}
	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(fma(t_0, Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sin(x) * sqrt(2.0))), 2.0) / fma(Float64(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0))
	tmp = 0.0
	if (x <= -0.043)
		tmp = t_2;
	elseif (x <= 1.55)
		tmp = Float64(fma(Float64(t_0 * sqrt(2.0)), Float64(Float64(sin(y) - Float64(fma(Float64(x * x), -0.010416666666666666, 0.0625) * x)) * fma(-0.0625, sin(y), sin(x))), 2.0) / fma(fma(t_1, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.5, 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.043], t$95$2, If[LessEqual[x, 1.55], N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[(N[(x * x), $MachinePrecision] * -0.010416666666666666 + 0.0625), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.042999999999999997 or 1.55 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \color{blue}{\left(\sin x \cdot \sqrt{2}\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x \cdot \color{blue}{\sqrt{2}}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x \cdot \sqrt{\color{blue}{2}}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-sqrt.f64 63.9%

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 63.9%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \color{blue}{\left(\sin x \cdot \sqrt{2}\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -0.042999999999999997 < x < 1.55

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.1%

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

   4. Applied rewrites 51.1%

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

   6. Applied rewrites 51.1%

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

Alternative 5: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 0.7639320225002103 \cdot \cos y\\ t_2 := \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right) \cdot 0.5, 3, 3\right)}\\ t_3 := 1 + -0.5 \cdot {x}^{2}\\ \mathbf{if}\;x \leq -0.043:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.045:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(t\_3 - \cos y\right)}{3 + \frac{\mathsf{fma}\left(t\_0, t\_3, t\_1\right)}{2} \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5.0) 1.0))
       (t_1 (* 0.7639320225002103 (cos y)))
       (t_2
        (/
         (fma
          (- (cos x) (cos y))
          (* (- (sin y) (* (sin x) 0.0625)) (* (sin x) (sqrt 2.0)))
          2.0)
         (fma (* (fma t_0 (cos x) t_1) 0.5) 3.0 3.0)))
       (t_3 (+ 1.0 (* -0.5 (pow x 2.0)))))
  (if (<= x -0.043)
    t_2
    (if (<= x 0.045)
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (- (sin y) (* 0.0625 x)))
         (- t_3 (cos y))))
       (+ 3.0 (* (/ (fma t_0 t_3 t_1) 2.0) 3.0)))
      t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 0.7639320225002103 * cos(y);
	double t_2 = fma((cos(x) - cos(y)), ((sin(y) - (sin(x) * 0.0625)) * (sin(x) * sqrt(2.0))), 2.0) / fma((fma(t_0, cos(x), t_1) * 0.5), 3.0, 3.0);
	double t_3 = 1.0 + (-0.5 * pow(x, 2.0));
	double tmp;
	if (x <= -0.043) {
		tmp = t_2;
	} else if (x <= 0.045) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * (t_3 - cos(y)))) / (3.0 + ((fma(t_0, t_3, t_1) / 2.0) * 3.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(0.7639320225002103 * cos(y))
	t_2 = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sin(x) * sqrt(2.0))), 2.0) / fma(Float64(fma(t_0, cos(x), t_1) * 0.5), 3.0, 3.0))
	t_3 = Float64(1.0 + Float64(-0.5 * (x ^ 2.0)))
	tmp = 0.0
	if (x <= -0.043)
		tmp = t_2;
	elseif (x <= 0.045)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * x))) * Float64(t_3 - cos(y)))) / Float64(3.0 + Float64(Float64(fma(t_0, t_3, t_1) / 2.0) * 3.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.043], t$95$2, If[LessEqual[x, 0.045], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$0 * t$95$3 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.042999999999999997 or 0.044999999999999998 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \color{blue}{\left(\sin x \cdot \sqrt{2}\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x \cdot \color{blue}{\sqrt{2}}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x \cdot \sqrt{\color{blue}{2}}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-sqrt.f64 63.9%

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 63.9%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \color{blue}{\left(\sin x \cdot \sqrt{2}\right)}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -0.042999999999999997 < x < 0.044999999999999998

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{\frac{1}{16} \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   6. Step-by-step derivation

      1. lower-*.f64 51.4%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   7. Applied rewrites 51.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{0.0625 \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{1}{16} \cdot x\right)\right) \cdot \left(\left(1 + \color{blue}{\frac{-1}{2} \cdot {x}^{2}}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{1}{16} \cdot x\right)\right) \cdot \left(\left(1 + \frac{-1}{2} \cdot \color{blue}{{x}^{2}}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      3. lower-pow.f64 51.0%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{\color{blue}{2}}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   10. Applied rewrites 51.0%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1 + \frac{-1}{2} \cdot {x}^{2}}, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{1}{16} \cdot x\right)\right) \cdot \left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1 + \color{blue}{\frac{-1}{2} \cdot {x}^{2}}, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{1}{16} \cdot x\right)\right) \cdot \left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1 + \frac{-1}{2} \cdot \color{blue}{{x}^{2}}, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

       3. lower-pow.f64 50.6%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1 + -0.5 \cdot {x}^{\color{blue}{2}}, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   13. Applied rewrites 50.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1 + -0.5 \cdot {x}^{2}}, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(t\_0, \sin y \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}\\ t_3 := \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y\\ \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sin x - t\_3 \cdot 0.0625\right) \cdot \sqrt{2}, \left(t\_3 - \sin x \cdot 0.0625\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot 0.5, 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) (cos y)))
       (t_1 (- (sqrt 5.0) 1.0))
       (t_2
        (/
         (fma
          t_0
          (* (sin y) (* (- (sin x) (* (sin y) 0.0625)) (sqrt 2.0)))
          2.0)
         (fma
          (* (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 0.5)
          3.0
          3.0)))
       (t_3 (* (fma (* y y) -0.16666666666666666 1.0) y)))
  (if (<= y -0.06)
    t_2
    (if (<= y 2.3)
      (/
       (fma
        (* (- (sin x) (* t_3 0.0625)) (sqrt 2.0))
        (* (- t_3 (* (sin x) 0.0625)) t_0)
        2.0)
       (fma
        (* (fma t_1 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 0.5)
        3.0
        3.0))
      t_2))))

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 = fma(t_0, (sin(y) * ((sin(x) - (sin(y) * 0.0625)) * sqrt(2.0))), 2.0) / fma((fma(t_1, cos(x), (0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0);
	double t_3 = fma((y * y), -0.16666666666666666, 1.0) * y;
	double tmp;
	if (y <= -0.06) {
		tmp = t_2;
	} else if (y <= 2.3) {
		tmp = fma(((sin(x) - (t_3 * 0.0625)) * sqrt(2.0)), ((t_3 - (sin(x) * 0.0625)) * t_0), 2.0) / fma((fma(t_1, cos(x), ((3.0 - sqrt(5.0)) * cos(y))) * 0.5), 3.0, 3.0);
	} else {
		tmp = t_2;
	}
	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(fma(t_0, Float64(sin(y) * Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * sqrt(2.0))), 2.0) / fma(Float64(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0))
	t_3 = Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y)
	tmp = 0.0
	if (y <= -0.06)
		tmp = t_2;
	elseif (y <= 2.3)
		tmp = Float64(fma(Float64(Float64(sin(x) - Float64(t_3 * 0.0625)) * sqrt(2.0)), Float64(Float64(t_3 - Float64(sin(x) * 0.0625)) * t_0), 2.0) / fma(Float64(fma(t_1, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) * 0.5), 3.0, 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -0.06], t$95$2, If[LessEqual[y, 2.3], N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(t$95$3 * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.059999999999999998 or 2.2999999999999998 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\sin y} \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-sin.f64 64.3%

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \sin y \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 64.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\sin y} \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -0.059999999999999998 < y < 2.2999999999999998

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.5%

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}{16}\right)\right) \cdot \left(\color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} \cdot 3} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.0%

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

   9. Applied rewrites 50.0%

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

   11. Applied rewrites 50.0%

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

Alternative 7: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := 1 - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y\\ t_3 := 0.7639320225002103 \cdot \cos y\\ \mathbf{if}\;y \leq -0.0275:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_3\right) \cdot 0.5, 3, 3\right)}\\ \mathbf{elif}\;y \leq 1.9:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sin x - t\_2 \cdot 0.0625\right) \cdot \sqrt{2}, \left(t\_2 - \sin x \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot 0.5, 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{3 + \frac{\mathsf{fma}\left(t\_1, 1, t\_3\right)}{2} \cdot 3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 1.0 (cos y)))
       (t_1 (- (sqrt 5.0) 1.0))
       (t_2 (* (fma (* y y) -0.16666666666666666 1.0) y))
       (t_3 (* 0.7639320225002103 (cos y))))
  (if (<= y -0.0275)
    (/
     (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) t_0))))
     (fma (* (fma t_1 (cos x) t_3) 0.5) 3.0 3.0))
    (if (<= y 1.9)
      (/
       (fma
        (* (- (sin x) (* t_2 0.0625)) (sqrt 2.0))
        (* (- t_2 (* (sin x) 0.0625)) (- (cos x) (cos y)))
        2.0)
       (fma
        (* (fma t_1 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 0.5)
        3.0
        3.0))
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (- (sin y) (/ (sin x) 16.0)))
         t_0))
       (+ 3.0 (* (/ (fma t_1 1.0 t_3) 2.0) 3.0)))))))

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 = fma((y * y), -0.16666666666666666, 1.0) * y;
	double t_3 = 0.7639320225002103 * cos(y);
	double tmp;
	if (y <= -0.0275) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * t_0)))) / fma((fma(t_1, cos(x), t_3) * 0.5), 3.0, 3.0);
	} else if (y <= 1.9) {
		tmp = fma(((sin(x) - (t_2 * 0.0625)) * sqrt(2.0)), ((t_2 - (sin(x) * 0.0625)) * (cos(x) - cos(y))), 2.0) / fma((fma(t_1, cos(x), ((3.0 - sqrt(5.0)) * cos(y))) * 0.5), 3.0, 3.0);
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * t_0)) / (3.0 + ((fma(t_1, 1.0, t_3) / 2.0) * 3.0));
	}
	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(fma(Float64(y * y), -0.16666666666666666, 1.0) * y)
	t_3 = Float64(0.7639320225002103 * cos(y))
	tmp = 0.0
	if (y <= -0.0275)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * t_0)))) / fma(Float64(fma(t_1, cos(x), t_3) * 0.5), 3.0, 3.0));
	elseif (y <= 1.9)
		tmp = Float64(fma(Float64(Float64(sin(x) - Float64(t_2 * 0.0625)) * sqrt(2.0)), Float64(Float64(t_2 - Float64(sin(x) * 0.0625)) * Float64(cos(x) - cos(y))), 2.0) / fma(Float64(fma(t_1, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) * 0.5), 3.0, 3.0));
	else
		tmp = 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))) * t_0)) / Float64(3.0 + Float64(Float64(fma(t_1, 1.0, t_3) / 2.0) * 3.0)));
	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[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$3 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0275], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9], N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(t$95$2 * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$1 * 1.0 + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0275

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      9. lower-cos.f64 62.5%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -0.0275 < y < 1.8999999999999999

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.5%

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}{16}\right)\right) \cdot \left(\color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} \cdot 3} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.0%

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

   9. Applied rewrites 50.0%

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

   11. Applied rewrites 50.0%

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

   if 1.8999999999999999 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{1} - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{1} - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := 0.7639320225002103 \cdot \cos y\\ t_1 := 1 + -0.5 \cdot {x}^{2}\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, t\_0\right) \cdot 0.5, 3, 3\right)}\\ \mathbf{if}\;x \leq -0.046:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.7:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 + \frac{\mathsf{fma}\left(t\_2, t\_1, t\_0\right)}{2} \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 0.7639320225002103 (cos y)))
       (t_1 (+ 1.0 (* -0.5 (pow x 2.0))))
       (t_2 (- (sqrt 5.0) 1.0))
       (t_3
        (/
         (+
          2.0
          (*
           -0.0625
           (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (fma (* (fma t_2 (cos x) t_0) 0.5) 3.0 3.0))))
  (if (<= x -0.046)
    t_3
    (if (<= x 1.7)
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (- (sin y) (* 0.0625 x)))
         (- t_1 (cos y))))
       (+ 3.0 (* (/ (fma t_2 t_1 t_0) 2.0) 3.0)))
      t_3))))

C

double code(double x, double y) {
	double t_0 = 0.7639320225002103 * cos(y);
	double t_1 = 1.0 + (-0.5 * pow(x, 2.0));
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma((fma(t_2, cos(x), t_0) * 0.5), 3.0, 3.0);
	double tmp;
	if (x <= -0.046) {
		tmp = t_3;
	} else if (x <= 1.7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * (t_1 - cos(y)))) / (3.0 + ((fma(t_2, t_1, t_0) / 2.0) * 3.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.7639320225002103 * cos(y))
	t_1 = Float64(1.0 + Float64(-0.5 * (x ^ 2.0)))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(Float64(fma(t_2, cos(x), t_0) * 0.5), 3.0, 3.0))
	tmp = 0.0
	if (x <= -0.046)
		tmp = t_3;
	elseif (x <= 1.7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * x))) * Float64(t_1 - cos(y)))) / Float64(3.0 + Float64(Float64(fma(t_2, t_1, t_0) / 2.0) * 3.0)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.046], t$95$3, If[LessEqual[x, 1.7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$2 * t$95$1 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.045999999999999999 or 1.7 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      9. lower-cos.f64 62.2%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 62.2%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -0.045999999999999999 < x < 1.7

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{\frac{1}{16} \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   6. Step-by-step derivation

      1. lower-*.f64 51.4%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   7. Applied rewrites 51.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{0.0625 \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{1}{16} \cdot x\right)\right) \cdot \left(\left(1 + \color{blue}{\frac{-1}{2} \cdot {x}^{2}}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{1}{16} \cdot x\right)\right) \cdot \left(\left(1 + \frac{-1}{2} \cdot \color{blue}{{x}^{2}}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      3. lower-pow.f64 51.0%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{\color{blue}{2}}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   10. Applied rewrites 51.0%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1 + \frac{-1}{2} \cdot {x}^{2}}, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{1}{16} \cdot x\right)\right) \cdot \left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1 + \color{blue}{\frac{-1}{2} \cdot {x}^{2}}, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{1}{16} \cdot x\right)\right) \cdot \left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1 + \frac{-1}{2} \cdot \color{blue}{{x}^{2}}, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

       3. lower-pow.f64 50.6%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, 1 + -0.5 \cdot {x}^{\color{blue}{2}}, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   13. Applied rewrites 50.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1 + -0.5 \cdot {x}^{2}}, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)\\ t_1 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(t\_0 \cdot 0.5, 3, 3\right)}\\ \mathbf{if}\;x \leq -0.046:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.72:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(1 - \cos y\right)}{3 + \frac{t\_0}{2} \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0
        (fma
         (- (sqrt 5.0) 1.0)
         (cos x)
         (* 0.7639320225002103 (cos y))))
       (t_1
        (/
         (+
          2.0
          (*
           -0.0625
           (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (fma (* t_0 0.5) 3.0 3.0))))
  (if (<= x -0.046)
    t_1
    (if (<= x 0.72)
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (- (sin y) (* 0.0625 x)))
         (- 1.0 (cos y))))
       (+ 3.0 (* (/ t_0 2.0) 3.0)))
      t_1))))

C

double code(double x, double y) {
	double t_0 = fma((sqrt(5.0) - 1.0), cos(x), (0.7639320225002103 * cos(y)));
	double t_1 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma((t_0 * 0.5), 3.0, 3.0);
	double tmp;
	if (x <= -0.046) {
		tmp = t_1;
	} else if (x <= 0.72) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * (1.0 - cos(y)))) / (3.0 + ((t_0 / 2.0) * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(0.7639320225002103 * cos(y)))
	t_1 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(Float64(t_0 * 0.5), 3.0, 3.0))
	tmp = 0.0
	if (x <= -0.046)
		tmp = t_1;
	elseif (x <= 0.72)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * x))) * Float64(1.0 - cos(y)))) / Float64(3.0 + Float64(Float64(t_0 / 2.0) * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.046], t$95$1, If[LessEqual[x, 0.72], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.045999999999999999 or 0.71999999999999997 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      9. lower-cos.f64 62.2%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 62.2%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -0.045999999999999999 < x < 0.71999999999999997

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{\frac{1}{16} \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   6. Step-by-step derivation

      1. lower-*.f64 51.4%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   7. Applied rewrites 51.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{0.0625 \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \color{blue}{\left(1 - \cos y\right)}}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{1}{16} \cdot x\right)\right) \cdot \left(1 - \color{blue}{\cos y}\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      2. lower-cos.f64 57.7%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(1 - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   10. Applied rewrites 57.7%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \color{blue}{\left(1 - \cos y\right)}}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := 0.7639320225002103 \cdot \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_0\right) \cdot 0.5, 3, 3\right)}\\ \mathbf{if}\;x \leq -140:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.29:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, 1, t\_0\right) \cdot 0.5, 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 0.7639320225002103 (cos y)))
       (t_1 (- (sqrt 5.0) 1.0))
       (t_2
        (/
         (+
          2.0
          (*
           -0.0625
           (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (fma (* (fma t_1 (cos x) t_0) 0.5) 3.0 3.0))))
  (if (<= x -140.0)
    t_2
    (if (<= x 0.29)
      (/
       (fma
        (- 1.0 (cos y))
        (*
         (- (sin y) (* (sin x) 0.0625))
         (* (- (sin x) (* (sin y) 0.0625)) (sqrt 2.0)))
        2.0)
       (fma (* (fma t_1 1.0 t_0) 0.5) 3.0 3.0))
      t_2))))

C

double code(double x, double y) {
	double t_0 = 0.7639320225002103 * cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma((fma(t_1, cos(x), t_0) * 0.5), 3.0, 3.0);
	double tmp;
	if (x <= -140.0) {
		tmp = t_2;
	} else if (x <= 0.29) {
		tmp = fma((1.0 - cos(y)), ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (sin(y) * 0.0625)) * sqrt(2.0))), 2.0) / fma((fma(t_1, 1.0, t_0) * 0.5), 3.0, 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.7639320225002103 * cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(Float64(fma(t_1, cos(x), t_0) * 0.5), 3.0, 3.0))
	tmp = 0.0
	if (x <= -140.0)
		tmp = t_2;
	elseif (x <= 0.29)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * sqrt(2.0))), 2.0) / fma(Float64(fma(t_1, 1.0, t_0) * 0.5), 3.0, 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -140.0], t$95$2, If[LessEqual[x, 0.29], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * 1.0 + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -140 or 0.28999999999999998 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      9. lower-cos.f64 62.2%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 62.2%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -140 < x < 0.28999999999999998

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{1} - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{1} - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := 0.7639320225002103 \cdot \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_0\right) \cdot 0.5, 3, 3\right)}\\ \mathbf{if}\;x \leq -140:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.29:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(1 - \cos y\right)}{3 + \frac{\mathsf{fma}\left(t\_1, 1, t\_0\right)}{2} \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 0.7639320225002103 (cos y)))
       (t_1 (- (sqrt 5.0) 1.0))
       (t_2
        (/
         (+
          2.0
          (*
           -0.0625
           (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (fma (* (fma t_1 (cos x) t_0) 0.5) 3.0 3.0))))
  (if (<= x -140.0)
    t_2
    (if (<= x 0.29)
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (- (sin y) (* 0.0625 x)))
         (- 1.0 (cos y))))
       (+ 3.0 (* (/ (fma t_1 1.0 t_0) 2.0) 3.0)))
      t_2))))

C

double code(double x, double y) {
	double t_0 = 0.7639320225002103 * cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma((fma(t_1, cos(x), t_0) * 0.5), 3.0, 3.0);
	double tmp;
	if (x <= -140.0) {
		tmp = t_2;
	} else if (x <= 0.29) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * (1.0 - cos(y)))) / (3.0 + ((fma(t_1, 1.0, t_0) / 2.0) * 3.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.7639320225002103 * cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(Float64(fma(t_1, cos(x), t_0) * 0.5), 3.0, 3.0))
	tmp = 0.0
	if (x <= -140.0)
		tmp = t_2;
	elseif (x <= 0.29)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * x))) * Float64(1.0 - cos(y)))) / Float64(3.0 + Float64(Float64(fma(t_1, 1.0, t_0) / 2.0) * 3.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -140.0], t$95$2, If[LessEqual[x, 0.29], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$1 * 1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -140 or 0.28999999999999998 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      9. lower-cos.f64 62.2%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 62.2%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -140 < x < 0.28999999999999998

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{\frac{1}{16} \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   6. Step-by-step derivation

      1. lower-*.f64 51.4%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   7. Applied rewrites 51.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{0.0625 \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\color{blue}{1} - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.7%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\color{blue}{1} - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(1 - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   12. Step-by-step derivation

   13. Applied rewrites 55.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(1 - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}\\ \mathbf{if}\;x \leq -140:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.29:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5.0) 1.0))
       (t_1
        (/
         (+
          2.0
          (*
           -0.0625
           (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (fma
          (* (fma t_0 (cos x) (* 0.7639320225002103 (cos y))) 0.5)
          3.0
          3.0))))
  (if (<= x -140.0)
    t_1
    (if (<= x 0.29)
      (/
       (+
        2.0
        (*
         (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y))
         (- 1.0 (cos y))))
       (*
        3.0
        (+ 1.0 (fma 0.5 (* (cos y) (- 3.0 (sqrt 5.0))) (* 0.5 t_0)))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma((fma(t_0, cos(x), (0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0);
	double tmp;
	if (x <= -140.0) {
		tmp = t_1;
	} else if (x <= 0.29) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * (1.0 - cos(y)))) / (3.0 * (1.0 + fma(0.5, (cos(y) * (3.0 - sqrt(5.0))), (0.5 * t_0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(Float64(fma(t_0, cos(x), Float64(0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0))
	tmp = 0.0
	if (x <= -140.0)
		tmp = t_1;
	elseif (x <= 0.29)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(y) * Float64(3.0 - sqrt(5.0))), Float64(0.5 * t_0)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -140.0], t$95$1, If[LessEqual[x, 0.29], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -140 or 0.28999999999999998 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      9. lower-cos.f64 62.2%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 62.2%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -140 < x < 0.28999999999999998

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 60.0%

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-sin.f64 60.0%

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

   7. Applied rewrites 60.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.0%

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

Alternative 13: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)\\ t_2 := \mathsf{fma}\left(t\_0, \cos x, 0.7639320225002103 \cdot \cos y\right)\\ t_3 := 2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\\ \mathbf{if}\;y \leq -4400000:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(t\_2 \cdot 0.5, 3, 3\right)}\\ \mathbf{elif}\;y \leq 28000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t\_1}, 0.3333333333333333, \left(\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot -0.0625\right) \cdot \frac{\left(\cos x - 1\right) \cdot \sqrt{2}}{t\_1}\right) \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{3 + \frac{t\_2}{2} \cdot 3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5.0) 1.0))
       (t_1 (fma (fma t_0 (cos x) (- 3.0 (sqrt 5.0))) 0.5 1.0))
       (t_2 (fma t_0 (cos x) (* 0.7639320225002103 (cos y))))
       (t_3
        (+
         2.0
         (*
          -0.0625
          (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))))
  (if (<= y -4400000.0)
    (/ t_3 (fma (* t_2 0.5) 3.0 3.0))
    (if (<= y 28000000.0)
      (fma
       (/ 2.0 t_1)
       0.3333333333333333
       (*
        (*
         (* (- 0.5 (* 0.5 (cos (* 2.0 x)))) -0.0625)
         (/ (* (- (cos x) 1.0) (sqrt 2.0)) t_1))
        0.3333333333333333))
      (/ t_3 (+ 3.0 (* (/ t_2 2.0) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma(fma(t_0, cos(x), (3.0 - sqrt(5.0))), 0.5, 1.0);
	double t_2 = fma(t_0, cos(x), (0.7639320225002103 * cos(y)));
	double t_3 = 2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))));
	double tmp;
	if (y <= -4400000.0) {
		tmp = t_3 / fma((t_2 * 0.5), 3.0, 3.0);
	} else if (y <= 28000000.0) {
		tmp = fma((2.0 / t_1), 0.3333333333333333, ((((0.5 - (0.5 * cos((2.0 * x)))) * -0.0625) * (((cos(x) - 1.0) * sqrt(2.0)) / t_1)) * 0.3333333333333333));
	} else {
		tmp = t_3 / (3.0 + ((t_2 / 2.0) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(fma(t_0, cos(x), Float64(3.0 - sqrt(5.0))), 0.5, 1.0)
	t_2 = fma(t_0, cos(x), Float64(0.7639320225002103 * cos(y)))
	t_3 = Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))))
	tmp = 0.0
	if (y <= -4400000.0)
		tmp = Float64(t_3 / fma(Float64(t_2 * 0.5), 3.0, 3.0));
	elseif (y <= 28000000.0)
		tmp = fma(Float64(2.0 / t_1), 0.3333333333333333, Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * -0.0625) * Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) / t_1)) * 0.3333333333333333));
	else
		tmp = Float64(t_3 / Float64(3.0 + Float64(Float64(t_2 / 2.0) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4400000.0], N[(t$95$3 / N[(N[(t$95$2 * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 28000000.0], N[(N[(2.0 / t$95$1), $MachinePrecision] * 0.3333333333333333 + N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(3.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4e6

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      9. lower-cos.f64 62.5%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -4.4e6 < y < 2.8e7

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 59.8%

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

   6. Applied rewrites 59.8%

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

   if 2.8e7 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right)}{2} \cdot 3} \]

      8. lower-cos.f64 62.5%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]

   7. Applied rewrites 62.5%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right)}{2} \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)\\ t_1 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{t\_0}\\ \mathbf{if}\;x \leq -1450000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 225000:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0
        (fma
         (*
          (fma
           (- (sqrt 5.0) 1.0)
           (cos x)
           (* 0.7639320225002103 (cos y)))
          0.5)
         3.0
         3.0))
       (t_1
        (/
         (+
          2.0
          (*
           -0.0625
           (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
         t_0)))
  (if (<= x -1450000.0)
    t_1
    (if (<= x 225000.0)
      (/
       (+
        2.0
        (*
         -0.0625
         (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       t_0)
      t_1))))

C

double code(double x, double y) {
	double t_0 = fma((fma((sqrt(5.0) - 1.0), cos(x), (0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0);
	double t_1 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / t_0;
	double tmp;
	if (x <= -1450000.0) {
		tmp = t_1;
	} else if (x <= 225000.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(0.7639320225002103 * cos(y))) * 0.5), 3.0, 3.0)
	t_1 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / t_0)
	tmp = 0.0
	if (x <= -1450000.0)
		tmp = t_1;
	elseif (x <= 225000.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -1450000.0], t$95$1, If[LessEqual[x, 225000.0], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e6 or 225000 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      9. lower-cos.f64 62.2%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 62.2%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -1.45e6 < x < 225000

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      9. lower-cos.f64 62.5%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := 0.7639320225002103 \cdot \cos y\\ t_1 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right) \cdot 0.5, 3, 3\right)}\\ \mathbf{if}\;x \leq -140:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.29:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + t\_0\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 0.7639320225002103 (cos y)))
       (t_1
        (/
         (+
          2.0
          (*
           -0.0625
           (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (fma (* (fma (- (sqrt 5.0) 1.0) (cos x) t_0) 0.5) 3.0 3.0))))
  (if (<= x -140.0)
    t_1
    (if (<= x 0.29)
      (/
       (+
        2.0
        (*
         -0.0625
         (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) t_0) 1.0))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = 0.7639320225002103 * cos(y);
	double t_1 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma((fma((sqrt(5.0) - 1.0), cos(x), t_0) * 0.5), 3.0, 3.0);
	double tmp;
	if (x <= -140.0) {
		tmp = t_1;
	} else if (x <= 0.29) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + t_0) - 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.7639320225002103 * cos(y))
	t_1 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), t_0) * 0.5), 3.0, 3.0))
	tmp = 0.0
	if (x <= -140.0)
		tmp = t_1;
	elseif (x <= 0.29)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + t_0) - 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -140.0], t$95$1, If[LessEqual[x, 0.29], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -140 or 0.28999999999999998 < x

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) \cdot \frac{1}{2}, 3, 3\right)} \]

      9. lower-cos.f64 62.2%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   9. Applied rewrites 62.2%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right) \cdot 0.5, 3, 3\right)} \]

   if -140 < x < 0.28999999999999998

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]

   7. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 78.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - 1\\ \mathbf{if}\;x \leq -140:\\ \;\;\;\;\frac{1}{\frac{3 + 1.5 \cdot \left(\left(3 + \cos x \cdot t\_0\right) - \sqrt{5}\right)}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}}\\ \mathbf{elif}\;x \leq 225000:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- (cos x) 1.0)))
  (if (<= x -140.0)
    (/
     1.0
     (/
      (+ 3.0 (* 1.5 (- (+ 3.0 (* (cos x) t_0)) (sqrt 5.0))))
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_1))))))
    (if (<= x 225000.0)
      (/
       (+
        2.0
        (*
         -0.0625
         (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+
        3.0
        (*
         1.5
         (- (+ (sqrt 5.0) (* 0.7639320225002103 (cos y))) 1.0))))
      (*
       (fma
        (* -0.0625 (* t_1 (sqrt 2.0)))
        (- 0.5 (* 0.5 (cos (* 2.0 x))))
        2.0)
       (*
        (/ 1.0 (fma (fma t_0 (cos x) (- 3.0 (sqrt 5.0))) 0.5 1.0))
        0.3333333333333333))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - 1.0;
	double tmp;
	if (x <= -140.0) {
		tmp = 1.0 / ((3.0 + (1.5 * ((3.0 + (cos(x) * t_0)) - sqrt(5.0)))) / (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_1)))));
	} else if (x <= 225000.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + (0.7639320225002103 * cos(y))) - 1.0)));
	} else {
		tmp = fma((-0.0625 * (t_1 * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) * ((1.0 / fma(fma(t_0, cos(x), (3.0 - sqrt(5.0))), 0.5, 1.0)) * 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - 1.0)
	tmp = 0.0
	if (x <= -140.0)
		tmp = Float64(1.0 / Float64(Float64(3.0 + Float64(1.5 * Float64(Float64(3.0 + Float64(cos(x) * t_0)) - sqrt(5.0)))) / Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_1))))));
	elseif (x <= 225000.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(0.7639320225002103 * cos(y))) - 1.0))));
	else
		tmp = Float64(fma(Float64(-0.0625 * Float64(t_1 * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0) * Float64(Float64(1.0 / fma(fma(t_0, cos(x), Float64(3.0 - sqrt(5.0))), 0.5, 1.0)) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -140.0], N[(1.0 / N[(N[(3.0 + N[(1.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 225000.0], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(1.0 / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -140

   1. Initial program 99.3%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 59.8%

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

   if -140 < x < 225000

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]

   7. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}} \]

   if 225000 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 59.8%

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

Alternative 17: 78.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333\right)\\ \mathbf{if}\;x \leq -140:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 225000:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0
        (*
         (fma
          (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x))))
          2.0)
         (*
          (/
           1.0
           (fma
            (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0)))
            0.5
            1.0))
          0.3333333333333333))))
  (if (<= x -140.0)
    t_0
    (if (<= x 225000.0)
      (/
       (+
        2.0
        (*
         -0.0625
         (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+
        3.0
        (*
         1.5
         (- (+ (sqrt 5.0) (* 0.7639320225002103 (cos y))) 1.0))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) * ((1.0 / fma(fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 0.5, 1.0)) * 0.3333333333333333);
	double tmp;
	if (x <= -140.0) {
		tmp = t_0;
	} else if (x <= 225000.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + (0.7639320225002103 * cos(y))) - 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0) * Float64(Float64(1.0 / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 0.5, 1.0)) * 0.3333333333333333))
	tmp = 0.0
	if (x <= -140.0)
		tmp = t_0;
	elseif (x <= 225000.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(0.7639320225002103 * cos(y))) - 1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(1.0 / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -140.0], t$95$0, If[LessEqual[x, 225000.0], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -140 or 225000 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 59.8%

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

   if -140 < x < 225000

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]

   7. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 78.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}\\ \mathbf{if}\;x \leq -140:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 225000:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0
        (/
         (*
          (fma
           (* 0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
           (- 0.5 (* 0.5 (cos (* 2.0 x))))
           -2.0)
          0.3333333333333333)
         (fma
          -0.5
          (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0)))
          -1.0))))
  (if (<= x -140.0)
    t_0
    (if (<= x 225000.0)
      (/
       (+
        2.0
        (*
         -0.0625
         (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+
        3.0
        (*
         1.5
         (- (+ (sqrt 5.0) (* 0.7639320225002103 (cos y))) 1.0))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = (fma((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), -1.0);
	double tmp;
	if (x <= -140.0) {
		tmp = t_0;
	} else if (x <= 225000.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + (0.7639320225002103 * cos(y))) - 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), -1.0))
	tmp = 0.0
	if (x <= -140.0)
		tmp = t_0;
	elseif (x <= 225000.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(0.7639320225002103 * cos(y))) - 1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -140.0], t$95$0, If[LessEqual[x, 225000.0], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -140 or 225000 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 59.8%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-PI.f64 43.2%

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

   6. Applied rewrites 43.2%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-PI.f64 40.8%

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

   8. Applied rewrites 40.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   10. Applied rewrites 43.2%

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

   12. Applied rewrites 59.8%

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

   if -140 < x < 225000

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right)}{2} \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\left(\sqrt{5} + \frac{6880887943736673}{9007199254740992} \cdot \cos y\right) - 1\right)}} \]

   7. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + 0.7639320225002103 \cdot \cos y\right) - 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 59.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 59.8%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-PI.f64 43.2%

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

6. Applied rewrites 43.2%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-PI.f64 40.8%

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

8. Applied rewrites 40.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

10. Applied rewrites 43.2%

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

12. Applied rewrites 59.8%

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

Alternative 20: 59.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 59.8%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-PI.f64 43.2%

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

6. Applied rewrites 43.2%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-PI.f64 40.8%

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

8. Applied rewrites 40.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

10. Applied rewrites 43.2%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 43.2%

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

12. Applied rewrites 59.8%

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

Alternative 21: 42.8% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 59.8%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-PI.f64 43.2%

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

6. Applied rewrites 43.2%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-PI.f64 40.8%

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

8. Applied rewrites 40.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

10. Applied rewrites 43.2%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 42.8%

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

Alternative 22: 40.2% accurate, 333.6× speedup

Math

\[0.3333333333333333 \]

FPCore

(FPCore (x y)
  :precision binary64
  0.3333333333333333)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 59.8%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-sqrt.f64 40.2%

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

7. Applied rewrites 40.2%

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

8. Evaluated real constant 40.2%

   \[\leadsto 0.3333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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))))))