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

Percentage Accurate: 99.3% → 99.3%

Time: 11.6s

Alternatives: 35

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: 99.3% accurate, 1.0× speedup

Math

\[\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot e^{0.34657359027997264}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (fma -0.0625 (sin y) (sin x)) (exp 0.34657359027997264))
   (* (fma -0.0625 (sin x) (sin y)) (- (cos x) (cos y)))
   2.0)
  (*
   3.0
   (+
    (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
    (* 0.38196601125010515 (cos y))))))

C

double code(double x, double y) {
	return fma((fma(-0.0625, sin(y), sin(x)) * exp(0.34657359027997264)), (fma(-0.0625, sin(x), sin(y)) * (cos(x) - cos(y))), 2.0) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (0.38196601125010515 * cos(y))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * exp(0.34657359027997264)), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(cos(x) - cos(y))), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(0.38196601125010515 * cos(y)))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Exp[0.34657359027997264], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Evaluated real constant 99.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. pow-to-exp N/A

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

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

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

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

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

   6. lower-unsound-log.f64 99.3%

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

6. Applied rewrites 99.3%

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

7. Evaluated real constant 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot e^{\color{blue}{0.34657359027997264}}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
8. Add Preprocessing

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0))
   (* (fma -0.0625 (sin x) (sin y)) (- (cos x) (cos y)))
   2.0)
  (*
   3.0
   (+
    (+ 1.0 (* 0.6180339887498949 (cos x)))
    (* 0.38196601125010515 (cos y))))))

C

double code(double x, double y) {
	return fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), (fma(-0.0625, sin(x), sin(y)) * (cos(x) - cos(y))), 2.0) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (0.38196601125010515 * cos(y))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(cos(x) - cos(y))), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(0.38196601125010515 * cos(y)))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Evaluated real constant 99.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Evaluated real constant 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
6. Add Preprocessing

Alternative 3: 99.3% accurate, 1.1× speedup

Math

\[\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(1 + \mathsf{fma}\left(0.38196601125010515, \cos y, 0.6180339887498949 \cdot \cos x\right)\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0))
   (* (fma -0.0625 (sin x) (sin y)) (- (cos x) (cos y)))
   2.0)
  (*
   3.0
   (+ 1.0 (fma 0.38196601125010515 (cos y) (* 0.6180339887498949 (cos x)))))))

C

double code(double x, double y) {
	return fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), (fma(-0.0625, sin(x), sin(y)) * (cos(x) - cos(y))), 2.0) / (3.0 * (1.0 + fma(0.38196601125010515, cos(y), (0.6180339887498949 * cos(x)))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(cos(x) - cos(y))), 2.0) / Float64(3.0 * Float64(1.0 + fma(0.38196601125010515, cos(y), Float64(0.6180339887498949 * cos(x))))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision] + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Evaluated real constant 99.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Evaluated real constant 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{6880887943736673}{18014398509481984} \cdot \cos y + \frac{347922205179541}{562949953421312} \cdot \cos x\right)\right)}} \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(1 + \mathsf{fma}\left(0.38196601125010515, \cos y, 0.6180339887498949 \cdot \cos x\right)\right)} \]

8. Applied rewrites 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(0.38196601125010515, \cos y, 0.6180339887498949 \cdot \cos x\right)\right)}} \]
9. Add Preprocessing

Alternative 4: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Evaluated real constant 99.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Evaluated real constant 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos y, \color{blue}{\frac{20642663831210019}{18014398509481984}}, \left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) \cdot 3\right)} \]

   10. lower-*.f64 99.3%

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

   11. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{20642663831210019}{18014398509481984}, \color{blue}{\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right)} \cdot 3\right)} \]

   12. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{20642663831210019}{18014398509481984}, \color{blue}{\left(\frac{347922205179541}{562949953421312} \cdot \cos x + 1\right)} \cdot 3\right)} \]

   13. lift-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{20642663831210019}{18014398509481984}, \left(\frac{347922205179541}{562949953421312} \cdot \color{blue}{\cos x} + 1\right) \cdot 3\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{20642663831210019}{18014398509481984}, \left(\color{blue}{\frac{347922205179541}{562949953421312} \cdot \cos x} + 1\right) \cdot 3\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{20642663831210019}{18014398509481984}, \left(\color{blue}{\cos x \cdot \frac{347922205179541}{562949953421312}} + 1\right) \cdot 3\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{20642663831210019}{18014398509481984}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{347922205179541}{562949953421312}, 1\right)} \cdot 3\right)} \]

   17. lift-cos.f64 99.3%

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

7. Applied rewrites 99.3%

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

Alternative 5: 99.3% accurate, 1.1× speedup

Math

\[\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, \left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y, 0.38196601125010515, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (*
   (fma
    (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0))
    (* (- (cos x) (cos y)) (fma (sin x) -0.0625 (sin y)))
    2.0)
   0.3333333333333333)
  (fma (cos y) 0.38196601125010515 (fma (cos x) 0.6180339887498949 1.0))))

C

double code(double x, double y) {
	return (fma((fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), ((cos(x) - cos(y)) * fma(sin(x), -0.0625, sin(y))), 2.0) * 0.3333333333333333) / fma(cos(y), 0.38196601125010515, fma(cos(x), 0.6180339887498949, 1.0));
}

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), Float64(Float64(cos(x) - cos(y)) * fma(sin(x), -0.0625, sin(y))), 2.0) * 0.3333333333333333) / fma(cos(y), 0.38196601125010515, fma(cos(x), 0.6180339887498949, 1.0)))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * 0.38196601125010515 + N[(N[Cos[x], $MachinePrecision] * 0.6180339887498949 + 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. Evaluated real constant 99.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Evaluated real constant 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

7. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, \left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos y, 0.38196601125010515, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right)}} \]
8. Add Preprocessing

Alternative 6: 99.3% accurate, 1.1× speedup

Math

\[\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, \left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, 0.38196601125010515, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (fma
   (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0))
   (* (- (cos x) (cos y)) (fma (sin x) -0.0625 (sin y)))
   2.0)
  (/
   0.3333333333333333
   (fma (cos y) 0.38196601125010515 (fma (cos x) 0.6180339887498949 1.0)))))

C

double code(double x, double y) {
	return fma((fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), ((cos(x) - cos(y)) * fma(sin(x), -0.0625, sin(y))), 2.0) * (0.3333333333333333 / fma(cos(y), 0.38196601125010515, fma(cos(x), 0.6180339887498949, 1.0)));
}

Julia

function code(x, y)
	return Float64(fma(Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), Float64(Float64(cos(x) - cos(y)) * fma(sin(x), -0.0625, sin(y))), 2.0) * Float64(0.3333333333333333 / fma(cos(y), 0.38196601125010515, fma(cos(x), 0.6180339887498949, 1.0))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[y], $MachinePrecision] * 0.38196601125010515 + N[(N[Cos[x], $MachinePrecision] * 0.6180339887498949 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Evaluated real constant 99.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Evaluated real constant 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

7. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, \left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, 0.38196601125010515, \mathsf{fma}\left(\cos x, 0.6180339887498949, 1\right)\right)}} \]
8. Add Preprocessing

Alternative 7: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 0.38196601125010515 \cdot \cos y\\ t_2 := \frac{\mathsf{fma}\left(\sin x \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_0, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t\_1\right)}\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \left(\sin y + x \cdot \left(0.010416666666666666 \cdot {x}^{2} - 0.0625\right)\right) \cdot t\_0, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* 0.38196601125010515 (cos y)))
        (t_2
         (/
          (fma
           (* (sin x) (sqrt 2.0))
           (* (fma -0.0625 (sin x) (sin y)) t_0)
           2.0)
          (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) t_1)))))
   (if (<= x -0.034)
     t_2
     (if (<= x 0.018)
       (/
        (fma
         (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0))
         (*
          (+ (sin y) (* x (- (* 0.010416666666666666 (pow x 2.0)) 0.0625)))
          t_0)
         2.0)
        (* 3.0 (+ (+ 1.0 (* 0.6180339887498949 (cos x))) t_1)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 0.38196601125010515 * cos(y);
	double t_2 = fma((sin(x) * sqrt(2.0)), (fma(-0.0625, sin(x), sin(y)) * t_0), 2.0) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_1));
	double tmp;
	if (x <= -0.034) {
		tmp = t_2;
	} else if (x <= 0.018) {
		tmp = fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), ((sin(y) + (x * ((0.010416666666666666 * pow(x, 2.0)) - 0.0625))) * t_0), 2.0) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(0.38196601125010515 * cos(y))
	t_2 = Float64(fma(Float64(sin(x) * sqrt(2.0)), Float64(fma(-0.0625, sin(x), sin(y)) * t_0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_1)))
	tmp = 0.0
	if (x <= -0.034)
		tmp = t_2;
	elseif (x <= 0.018)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), Float64(Float64(sin(y) + Float64(x * Float64(Float64(0.010416666666666666 * (x ^ 2.0)) - 0.0625))) * t_0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.034], t$95$2, If[LessEqual[x, 0.018], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(x * N[(N[(0.010416666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.034000000000000002 or 0.017999999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \sqrt{2}}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.8%

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

   7. Applied rewrites 64.8%

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

   if -0.034000000000000002 < x < 0.017999999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\left(\sin y + x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)\right)} \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \left(\sin y + \color{blue}{x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \left(\sin y + \color{blue}{x} \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \left(\sin y + x \cdot \color{blue}{\left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \left(\sin y + x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \color{blue}{\frac{1}{16}}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \left(\sin y + x \cdot \left(\frac{1}{96} \cdot {x}^{2} - \frac{1}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{347922205179541}{562949953421312} \cdot \cos x\right) + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)} \]

      6. lower-pow.f64 50.3%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \left(\sin y + x \cdot \left(0.010416666666666666 \cdot {x}^{2} - 0.0625\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\left(\sin y + x \cdot \left(0.010416666666666666 \cdot {x}^{2} - 0.0625\right)\right)} \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\\ t_2 := 0.38196601125010515 \cdot \cos y\\ t_3 := \frac{\mathsf{fma}\left(\sin x \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_0, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t\_2\right)}\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, t\_1\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, t\_1, \sin y\right) \cdot t\_0, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0)))))
        (t_2 (* 0.38196601125010515 (cos y)))
        (t_3
         (/
          (fma
           (* (sin x) (sqrt 2.0))
           (* (fma -0.0625 (sin x) (sin y)) t_0)
           2.0)
          (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) t_2)))))
   (if (<= x -0.034)
     t_3
     (if (<= x 0.018)
       (/
        (fma
         (* (fma -0.0625 (sin y) t_1) (sqrt 2.0))
         (* (fma -0.0625 t_1 (sin y)) t_0)
         2.0)
        (* 3.0 (+ (+ 1.0 (* 0.6180339887498949 (cos x))) t_2)))
       t_3))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)));
	double t_2 = 0.38196601125010515 * cos(y);
	double t_3 = fma((sin(x) * sqrt(2.0)), (fma(-0.0625, sin(x), sin(y)) * t_0), 2.0) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_2));
	double tmp;
	if (x <= -0.034) {
		tmp = t_3;
	} else if (x <= 0.018) {
		tmp = fma((fma(-0.0625, sin(y), t_1) * sqrt(2.0)), (fma(-0.0625, t_1, sin(y)) * t_0), 2.0) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))))
	t_2 = Float64(0.38196601125010515 * cos(y))
	t_3 = Float64(fma(Float64(sin(x) * sqrt(2.0)), Float64(fma(-0.0625, sin(x), sin(y)) * t_0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_2)))
	tmp = 0.0
	if (x <= -0.034)
		tmp = t_3;
	elseif (x <= 0.018)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), t_1) * sqrt(2.0)), Float64(fma(-0.0625, t_1, sin(y)) * t_0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + t_2)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.034], t$95$3, If[LessEqual[x, 0.018], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * t$95$1 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.034000000000000002 or 0.017999999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \sqrt{2}}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.8%

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

   7. Applied rewrites 64.8%

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

   if -0.034000000000000002 < x < 0.017999999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.3%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.6%

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right), \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   11. Applied rewrites 50.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right)\\ t_1 := 0.38196601125010515 \cdot \cos y\\ t_2 := \frac{\mathsf{fma}\left(\sin x \cdot \sqrt{2}, t\_0, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t\_1\right)}\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, t\_0, 2\right)}{3 \cdot \left(\left(1.618033988749895 + -0.30901699437494745 \cdot {x}^{2}\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma -0.0625 (sin x) (sin y)) (- (cos x) (cos y))))
        (t_1 (* 0.38196601125010515 (cos y)))
        (t_2
         (/
          (fma (* (sin x) (sqrt 2.0)) t_0 2.0)
          (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) t_1)))))
   (if (<= x -0.034)
     t_2
     (if (<= x 0.018)
       (/
        (fma (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)) t_0 2.0)
        (*
         3.0
         (+ (+ 1.618033988749895 (* -0.30901699437494745 (pow x 2.0))) t_1)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, sin(x), sin(y)) * (cos(x) - cos(y));
	double t_1 = 0.38196601125010515 * cos(y);
	double t_2 = fma((sin(x) * sqrt(2.0)), t_0, 2.0) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_1));
	double tmp;
	if (x <= -0.034) {
		tmp = t_2;
	} else if (x <= 0.018) {
		tmp = fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), t_0, 2.0) / (3.0 * ((1.618033988749895 + (-0.30901699437494745 * pow(x, 2.0))) + t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(-0.0625, sin(x), sin(y)) * Float64(cos(x) - cos(y)))
	t_1 = Float64(0.38196601125010515 * cos(y))
	t_2 = Float64(fma(Float64(sin(x) * sqrt(2.0)), t_0, 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_1)))
	tmp = 0.0
	if (x <= -0.034)
		tmp = t_2;
	elseif (x <= 0.018)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), t_0, 2.0) / Float64(3.0 * Float64(Float64(1.618033988749895 + Float64(-0.30901699437494745 * (x ^ 2.0))) + t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.034], t$95$2, If[LessEqual[x, 0.018], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.618033988749895 + N[(-0.30901699437494745 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.034000000000000002 or 0.017999999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \sqrt{2}}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.8%

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

   7. Applied rewrites 64.8%

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

   if -0.034000000000000002 < x < 0.017999999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{910872158600853}{562949953421312} + \frac{-347922205179541}{1125899906842624} \cdot {x}^{2}\right)} + 0.38196601125010515 \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(\frac{910872158600853}{562949953421312} + \color{blue}{\frac{-347922205179541}{1125899906842624} \cdot {x}^{2}}\right) + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(\frac{910872158600853}{562949953421312} + \frac{-347922205179541}{1125899906842624} \cdot \color{blue}{{x}^{2}}\right) + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)} \]

      3. lower-pow.f64 50.4%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1.618033988749895 + -0.30901699437494745 \cdot {x}^{\color{blue}{2}}\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   8. Applied rewrites 50.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\color{blue}{\left(1.618033988749895 + -0.30901699437494745 \cdot {x}^{2}\right)} + 0.38196601125010515 \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right)\\ t_1 := 0.38196601125010515 \cdot \cos y\\ t_2 := \frac{\mathsf{fma}\left(\sin x \cdot \sqrt{2}, t\_0, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t\_1\right)}\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, t\_0, 2\right)}{\mathsf{fma}\left(-0.9270509831248424, {x}^{2}, 3 \cdot \left(1.618033988749895 + t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma -0.0625 (sin x) (sin y)) (- (cos x) (cos y))))
        (t_1 (* 0.38196601125010515 (cos y)))
        (t_2
         (/
          (fma (* (sin x) (sqrt 2.0)) t_0 2.0)
          (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) t_1)))))
   (if (<= x -0.034)
     t_2
     (if (<= x 0.018)
       (/
        (fma (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0)) t_0 2.0)
        (fma
         -0.9270509831248424
         (pow x 2.0)
         (* 3.0 (+ 1.618033988749895 t_1))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, sin(x), sin(y)) * (cos(x) - cos(y));
	double t_1 = 0.38196601125010515 * cos(y);
	double t_2 = fma((sin(x) * sqrt(2.0)), t_0, 2.0) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_1));
	double tmp;
	if (x <= -0.034) {
		tmp = t_2;
	} else if (x <= 0.018) {
		tmp = fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), t_0, 2.0) / fma(-0.9270509831248424, pow(x, 2.0), (3.0 * (1.618033988749895 + t_1)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(-0.0625, sin(x), sin(y)) * Float64(cos(x) - cos(y)))
	t_1 = Float64(0.38196601125010515 * cos(y))
	t_2 = Float64(fma(Float64(sin(x) * sqrt(2.0)), t_0, 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_1)))
	tmp = 0.0
	if (x <= -0.034)
		tmp = t_2;
	elseif (x <= 0.018)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), t_0, 2.0) / fma(-0.9270509831248424, (x ^ 2.0), Float64(3.0 * Float64(1.618033988749895 + t_1))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.034], t$95$2, If[LessEqual[x, 0.018], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(-0.9270509831248424 * N[Power[x, 2.0], $MachinePrecision] + N[(3.0 * N[(1.618033988749895 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.034000000000000002 or 0.017999999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \sqrt{2}}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.8%

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

   7. Applied rewrites 64.8%

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

   if -0.034000000000000002 < x < 0.017999999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\frac{-1043766615538623}{1125899906842624} \cdot {x}^{2} + 3 \cdot \left(\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)}} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{-1043766615538623}{1125899906842624}, \color{blue}{{x}^{2}}, 3 \cdot \left(\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{-1043766615538623}{1125899906842624}, {x}^{\color{blue}{2}}, 3 \cdot \left(\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{-1043766615538623}{1125899906842624}, {x}^{2}, 3 \cdot \left(\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{-1043766615538623}{1125899906842624}, {x}^{2}, 3 \cdot \left(\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{-1043766615538623}{1125899906842624}, {x}^{2}, 3 \cdot \left(\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y\right)\right)} \]

      6. lower-cos.f64 50.4%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(-0.9270509831248424, {x}^{2}, 3 \cdot \left(1.618033988749895 + 0.38196601125010515 \cdot \cos y\right)\right)} \]

   8. Applied rewrites 50.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(-0.9270509831248424, {x}^{2}, 3 \cdot \left(1.618033988749895 + 0.38196601125010515 \cdot \cos y\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 1 + -0.5 \cdot {x}^{2}\\ t_1 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ t_2 := 0.38196601125010515 \cdot \cos y\\ t_3 := \frac{\mathsf{fma}\left(\sin x \cdot \sqrt{2}, t\_1 \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t\_2\right)}\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, t\_1 \cdot \left(t\_0 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot t\_0\right) + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -0.5 (pow x 2.0))))
        (t_1 (fma -0.0625 (sin x) (sin y)))
        (t_2 (* 0.38196601125010515 (cos y)))
        (t_3
         (/
          (fma (* (sin x) (sqrt 2.0)) (* t_1 (- (cos x) (cos y))) 2.0)
          (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) t_2)))))
   (if (<= x -0.034)
     t_3
     (if (<= x 0.018)
       (/
        (fma
         (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0))
         (* t_1 (- t_0 (cos y)))
         2.0)
        (* 3.0 (+ (+ 1.0 (* 0.6180339887498949 t_0)) t_2)))
       t_3))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-0.5 * pow(x, 2.0));
	double t_1 = fma(-0.0625, sin(x), sin(y));
	double t_2 = 0.38196601125010515 * cos(y);
	double t_3 = fma((sin(x) * sqrt(2.0)), (t_1 * (cos(x) - cos(y))), 2.0) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_2));
	double tmp;
	if (x <= -0.034) {
		tmp = t_3;
	} else if (x <= 0.018) {
		tmp = fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), (t_1 * (t_0 - cos(y))), 2.0) / (3.0 * ((1.0 + (0.6180339887498949 * t_0)) + t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-0.5 * (x ^ 2.0)))
	t_1 = fma(-0.0625, sin(x), sin(y))
	t_2 = Float64(0.38196601125010515 * cos(y))
	t_3 = Float64(fma(Float64(sin(x) * sqrt(2.0)), Float64(t_1 * Float64(cos(x) - cos(y))), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_2)))
	tmp = 0.0
	if (x <= -0.034)
		tmp = t_3;
	elseif (x <= 0.018)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), Float64(t_1 * Float64(t_0 - cos(y))), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * t_0)) + t_2)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.034], t$95$3, If[LessEqual[x, 0.018], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(t$95$0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.034000000000000002 or 0.017999999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \sqrt{2}}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.8%

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

   7. Applied rewrites 64.8%

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

   if -0.034000000000000002 < x < 0.017999999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 50.8%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{\color{blue}{2}}\right) - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   8. Applied rewrites 50.8%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 50.4%

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \left(1 + -0.5 \cdot {x}^{\color{blue}{2}}\right)\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   11. Applied rewrites 50.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)}\right) + 0.38196601125010515 \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 0.38196601125010515 \cdot \cos y\\ t_2 := \frac{\mathsf{fma}\left(\sin x \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_0, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t\_1\right)}\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \left(\sin y + -0.0625 \cdot x\right) \cdot t\_0, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* 0.38196601125010515 (cos y)))
        (t_2
         (/
          (fma
           (* (sin x) (sqrt 2.0))
           (* (fma -0.0625 (sin x) (sin y)) t_0)
           2.0)
          (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) t_1)))))
   (if (<= x -0.034)
     t_2
     (if (<= x 0.018)
       (/
        (fma
         (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0))
         (* (+ (sin y) (* -0.0625 x)) t_0)
         2.0)
        (* 3.0 (+ (+ 1.0 (* 0.6180339887498949 (cos x))) t_1)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 0.38196601125010515 * cos(y);
	double t_2 = fma((sin(x) * sqrt(2.0)), (fma(-0.0625, sin(x), sin(y)) * t_0), 2.0) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_1));
	double tmp;
	if (x <= -0.034) {
		tmp = t_2;
	} else if (x <= 0.018) {
		tmp = fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), ((sin(y) + (-0.0625 * x)) * t_0), 2.0) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(0.38196601125010515 * cos(y))
	t_2 = Float64(fma(Float64(sin(x) * sqrt(2.0)), Float64(fma(-0.0625, sin(x), sin(y)) * t_0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_1)))
	tmp = 0.0
	if (x <= -0.034)
		tmp = t_2;
	elseif (x <= 0.018)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), Float64(Float64(sin(y) + Float64(-0.0625 * x)) * t_0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.034], t$95$2, If[LessEqual[x, 0.018], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(-0.0625 * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.034000000000000002 or 0.017999999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \sqrt{2}}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.8%

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

   7. Applied rewrites 64.8%

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

   if -0.034000000000000002 < x < 0.017999999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\left(\sin y + \frac{-1}{16} \cdot x\right)} \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 50.7%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \left(\sin y + -0.0625 \cdot \color{blue}{x}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   8. Applied rewrites 50.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\left(\sin y + -0.0625 \cdot x\right)} \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := t\_0 \cdot \sqrt{2}\\ t_2 := \frac{0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ t_3 := \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\\ \mathbf{if}\;y \leq -0.03:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3 \cdot \sqrt{2}, \sin y \cdot t\_0, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 2.2:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, y\right), 2\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot t\_3, \sin y, 2\right) \cdot t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* t_0 (sqrt 2.0)))
        (t_2
         (/
          0.3333333333333333
          (fma
           0.38196601125010515
           (cos y)
           (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
        (t_3 (fma -0.0625 (sin y) (sin x))))
   (if (<= y -0.03)
     (/
      (fma (* t_3 (sqrt 2.0)) (* (sin y) t_0) 2.0)
      (*
       3.0
       (+
        (+ 1.0 (* 0.6180339887498949 (cos x)))
        (* 0.38196601125010515 (cos y)))))
     (if (<= y 2.2)
       (*
        (fma (* t_1 (fma -0.0625 y (sin x))) (fma -0.0625 (sin x) y) 2.0)
        t_2)
       (* (fma (* t_1 t_3) (sin y) 2.0) t_2)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = t_0 * sqrt(2.0);
	double t_2 = 0.3333333333333333 / fma(0.38196601125010515, cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	double t_3 = fma(-0.0625, sin(y), sin(x));
	double tmp;
	if (y <= -0.03) {
		tmp = fma((t_3 * sqrt(2.0)), (sin(y) * t_0), 2.0) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (0.38196601125010515 * cos(y))));
	} else if (y <= 2.2) {
		tmp = fma((t_1 * fma(-0.0625, y, sin(x))), fma(-0.0625, sin(x), y), 2.0) * t_2;
	} else {
		tmp = fma((t_1 * t_3), sin(y), 2.0) * t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(t_0 * sqrt(2.0))
	t_2 = Float64(0.3333333333333333 / fma(0.38196601125010515, cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)))
	t_3 = fma(-0.0625, sin(y), sin(x))
	tmp = 0.0
	if (y <= -0.03)
		tmp = Float64(fma(Float64(t_3 * sqrt(2.0)), Float64(sin(y) * t_0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(0.38196601125010515 * cos(y)))));
	elseif (y <= 2.2)
		tmp = Float64(fma(Float64(t_1 * fma(-0.0625, y, sin(x))), fma(-0.0625, sin(x), y), 2.0) * t_2);
	else
		tmp = Float64(fma(Float64(t_1 * t_3), sin(y), 2.0) * 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[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 / N[(0.38196601125010515 * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.03], N[(N[(N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2], N[(N[(N[(t$95$1 * N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + y), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(t$95$1 * t$95$3), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.029999999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\sin y} \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-sin.f64 63.8%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \sin y \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   8. Applied rewrites 63.8%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\sin y} \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   if -0.029999999999999999 < y < 2.2000000000000002

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 51.4%

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

   if 2.2000000000000002 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-sin.f64 63.7%

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

   7. Applied rewrites 63.7%

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

Alternative 14: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \sin y \cdot t\_0, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.03:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, y\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (/
          (fma
           (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0))
           (* (sin y) t_0)
           2.0)
          (*
           3.0
           (+
            (+ 1.0 (* 0.6180339887498949 (cos x)))
            (* 0.38196601125010515 (cos y)))))))
   (if (<= y -0.03)
     t_1
     (if (<= y 2.2)
       (*
        (fma
         (* (* t_0 (sqrt 2.0)) (fma -0.0625 y (sin x)))
         (fma -0.0625 (sin x) y)
         2.0)
        (/
         0.3333333333333333
         (fma
          0.38196601125010515
          (cos y)
          (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), (sin(y) * t_0), 2.0) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (0.38196601125010515 * cos(y))));
	double tmp;
	if (y <= -0.03) {
		tmp = t_1;
	} else if (y <= 2.2) {
		tmp = fma(((t_0 * sqrt(2.0)) * fma(-0.0625, y, sin(x))), fma(-0.0625, sin(x), y), 2.0) * (0.3333333333333333 / fma(0.38196601125010515, cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), Float64(sin(y) * t_0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(0.38196601125010515 * cos(y)))))
	tmp = 0.0
	if (y <= -0.03)
		tmp = t_1;
	elseif (y <= 2.2)
		tmp = Float64(fma(Float64(Float64(t_0 * sqrt(2.0)) * fma(-0.0625, y, sin(x))), fma(-0.0625, sin(x), y), 2.0) * Float64(0.3333333333333333 / fma(0.38196601125010515, cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.03], t$95$1, If[LessEqual[y, 2.2], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + y), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(0.38196601125010515 * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.029999999999999999 or 2.2000000000000002 < y

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\sin y} \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-sin.f64 63.8%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \sin y \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   8. Applied rewrites 63.8%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\sin y} \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   if -0.029999999999999999 < y < 2.2000000000000002

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 51.4%

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

Alternative 15: 80.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos x - 1\\ \mathbf{if}\;x \leq -0.035:\\ \;\;\;\;\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot t\_0\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0305:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(0.38196601125010515, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, -0.0625 \cdot \left(\sin x \cdot t\_0\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0)))
   (if (<= x -0.035)
     (*
      (fma (* (sin x) (* (sqrt 2.0) t_0)) (fma -0.0625 (sin x) (sin y)) 2.0)
      (/
       0.3333333333333333
       (fma
        0.38196601125010515
        (cos y)
        (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
     (if (<= x 0.0305)
       (/
        (/
         (fma
          (* (fma (sin y) -0.0625 x) (sqrt 2.0))
          (* (fma x -0.0625 (sin y)) (- (cos x) (cos y)))
          2.0)
         (fma
          0.38196601125010515
          (cos y)
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0)))
        3.0)
       (/
        (fma
         (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0))
         (* -0.0625 (* (sin x) t_0))
         2.0)
        (*
         3.0
         (+
          (+ 1.0 (* 0.6180339887498949 (cos x)))
          (* 0.38196601125010515 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double tmp;
	if (x <= -0.035) {
		tmp = fma((sin(x) * (sqrt(2.0) * t_0)), fma(-0.0625, sin(x), sin(y)), 2.0) * (0.3333333333333333 / fma(0.38196601125010515, cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	} else if (x <= 0.0305) {
		tmp = (fma((fma(sin(y), -0.0625, x) * sqrt(2.0)), (fma(x, -0.0625, sin(y)) * (cos(x) - cos(y))), 2.0) / fma(0.38196601125010515, cos(y), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0))) / 3.0;
	} else {
		tmp = fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), (-0.0625 * (sin(x) * t_0)), 2.0) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (0.38196601125010515 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	tmp = 0.0
	if (x <= -0.035)
		tmp = Float64(fma(Float64(sin(x) * Float64(sqrt(2.0) * t_0)), fma(-0.0625, sin(x), sin(y)), 2.0) * Float64(0.3333333333333333 / fma(0.38196601125010515, cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))));
	elseif (x <= 0.0305)
		tmp = Float64(Float64(fma(Float64(fma(sin(y), -0.0625, x) * sqrt(2.0)), Float64(fma(x, -0.0625, sin(y)) * Float64(cos(x) - cos(y))), 2.0) / fma(0.38196601125010515, cos(y), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0))) / 3.0);
	else
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), Float64(-0.0625 * Float64(sin(x) * t_0)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(0.38196601125010515 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.035], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(0.38196601125010515 * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0305], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(x * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.38196601125010515 * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.035000000000000003

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 63.2%

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

   7. Applied rewrites 63.2%

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

   if -0.035000000000000003 < x < 0.030499999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

   12. Applied rewrites 50.7%

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

   if 0.030499999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, -0.0625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 80.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)\\ t_1 := \cos x - 1\\ \mathbf{if}\;x \leq -0.035:\\ \;\;\;\;\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot t\_1\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.38196601125010515, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0305:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, -0.0625 \cdot \left(\sin x \cdot t\_1\right), 2\right)}{t\_0}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* 0.6180339887498949 (cos x)))
           (* 0.38196601125010515 (cos y)))))
        (t_1 (- (cos x) 1.0)))
   (if (<= x -0.035)
     (*
      (fma (* (sin x) (* (sqrt 2.0) t_1)) (fma -0.0625 (sin x) (sin y)) 2.0)
      (/
       0.3333333333333333
       (fma
        0.38196601125010515
        (cos y)
        (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
     (if (<= x 0.0305)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (sqrt 2.0))
         (* (fma -0.0625 x (sin y)) (- (cos x) (cos y)))
         2.0)
        t_0)
       (/
        (fma
         (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0))
         (* -0.0625 (* (sin x) t_1))
         2.0)
        t_0)))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + (0.38196601125010515 * cos(y)));
	double t_1 = cos(x) - 1.0;
	double tmp;
	if (x <= -0.035) {
		tmp = fma((sin(x) * (sqrt(2.0) * t_1)), fma(-0.0625, sin(x), sin(y)), 2.0) * (0.3333333333333333 / fma(0.38196601125010515, cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	} else if (x <= 0.0305) {
		tmp = fma((fma(-0.0625, sin(y), x) * sqrt(2.0)), (fma(-0.0625, x, sin(y)) * (cos(x) - cos(y))), 2.0) / t_0;
	} else {
		tmp = fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), (-0.0625 * (sin(x) * t_1)), 2.0) / t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + Float64(0.38196601125010515 * cos(y))))
	t_1 = Float64(cos(x) - 1.0)
	tmp = 0.0
	if (x <= -0.035)
		tmp = Float64(fma(Float64(sin(x) * Float64(sqrt(2.0) * t_1)), fma(-0.0625, sin(x), sin(y)), 2.0) * Float64(0.3333333333333333 / fma(0.38196601125010515, cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))));
	elseif (x <= 0.0305)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * sqrt(2.0)), Float64(fma(-0.0625, x, sin(y)) * Float64(cos(x) - cos(y))), 2.0) / t_0);
	else
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), Float64(-0.0625 * Float64(sin(x) * t_1)), 2.0) / t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.035], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(0.38196601125010515 * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0305], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sin[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.035000000000000003

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 63.2%

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

   7. Applied rewrites 63.2%

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

   if -0.035000000000000003 < x < 0.030499999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.7%

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

   11. Evaluated real constant 50.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   if 0.030499999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, -0.0625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 80.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 0.38196601125010515 \cdot \cos y\\ t_2 := 3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + t\_1\right)\\ \mathbf{if}\;x \leq -0.035:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t\_1\right)}\\ \mathbf{elif}\;x \leq 0.0305:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot t\_0, 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, -0.0625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right), 2\right)}{t\_2}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* 0.38196601125010515 (cos y)))
        (t_2 (* 3.0 (+ (+ 1.0 (* 0.6180339887498949 (cos x))) t_1))))
   (if (<= x -0.035)
     (/
      (+ 2.0 (* (* -0.0625 (* (pow (sin x) 2.0) (sqrt 2.0))) t_0))
      (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) t_1)))
     (if (<= x 0.0305)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (sqrt 2.0))
         (* (fma -0.0625 x (sin y)) t_0)
         2.0)
        t_2)
       (/
        (fma
         (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0))
         (* -0.0625 (* (sin x) (- (cos x) 1.0)))
         2.0)
        t_2)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 0.38196601125010515 * cos(y);
	double t_2 = 3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + t_1);
	double tmp;
	if (x <= -0.035) {
		tmp = (2.0 + ((-0.0625 * (pow(sin(x), 2.0) * sqrt(2.0))) * t_0)) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_1));
	} else if (x <= 0.0305) {
		tmp = fma((fma(-0.0625, sin(y), x) * sqrt(2.0)), (fma(-0.0625, x, sin(y)) * t_0), 2.0) / t_2;
	} else {
		tmp = fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), (-0.0625 * (sin(x) * (cos(x) - 1.0))), 2.0) / t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(0.38196601125010515 * cos(y))
	t_2 = Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + t_1))
	tmp = 0.0
	if (x <= -0.035)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * sqrt(2.0))) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_1)));
	elseif (x <= 0.0305)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * sqrt(2.0)), Float64(fma(-0.0625, x, sin(y)) * t_0), 2.0) / t_2);
	else
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), Float64(-0.0625 * Float64(sin(x) * Float64(cos(x) - 1.0))), 2.0) / 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[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.035], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0305], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.035000000000000003

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 63.1%

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

   5. Applied rewrites 63.1%

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

   if -0.035000000000000003 < x < 0.030499999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.7%

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

   11. Evaluated real constant 50.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   if 0.030499999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, -0.0625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 80.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := 0.38196601125010515 \cdot \cos y\\ t_1 := 3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t\_0\right)\\ \mathbf{if}\;x \leq -0.035:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{t\_1}\\ \mathbf{elif}\;x \leq 0.0305:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right), 2\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, -0.0625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.38196601125010515 (cos y)))
        (t_1 (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) t_0))))
   (if (<= x -0.035)
     (/
      (+
       2.0
       (* (* -0.0625 (* (pow (sin x) 2.0) (sqrt 2.0))) (- (cos x) (cos y))))
      t_1)
     (if (<= x 0.0305)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (sqrt 2.0))
         (* (fma -0.0625 x (sin y)) (- (+ 1.0 (* -0.5 (pow x 2.0))) (cos y)))
         2.0)
        t_1)
       (/
        (fma
         (* (fma -0.0625 (sin y) (sin x)) (sqrt 2.0))
         (* -0.0625 (* (sin x) (- (cos x) 1.0)))
         2.0)
        (* 3.0 (+ (+ 1.0 (* 0.6180339887498949 (cos x))) t_0)))))))

C

double code(double x, double y) {
	double t_0 = 0.38196601125010515 * cos(y);
	double t_1 = 3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_0);
	double tmp;
	if (x <= -0.035) {
		tmp = (2.0 + ((-0.0625 * (pow(sin(x), 2.0) * sqrt(2.0))) * (cos(x) - cos(y)))) / t_1;
	} else if (x <= 0.0305) {
		tmp = fma((fma(-0.0625, sin(y), x) * sqrt(2.0)), (fma(-0.0625, x, sin(y)) * ((1.0 + (-0.5 * pow(x, 2.0))) - cos(y))), 2.0) / t_1;
	} else {
		tmp = fma((fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), (-0.0625 * (sin(x) * (cos(x) - 1.0))), 2.0) / (3.0 * ((1.0 + (0.6180339887498949 * cos(x))) + t_0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.38196601125010515 * cos(y))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + t_0))
	tmp = 0.0
	if (x <= -0.035)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * sqrt(2.0))) * Float64(cos(x) - cos(y)))) / t_1);
	elseif (x <= 0.0305)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * sqrt(2.0)), Float64(fma(-0.0625, x, sin(y)) * Float64(Float64(1.0 + Float64(-0.5 * (x ^ 2.0))) - cos(y))), 2.0) / t_1);
	else
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), sin(x)) * sqrt(2.0)), Float64(-0.0625 * Float64(sin(x) * Float64(cos(x) - 1.0))), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(0.6180339887498949 * cos(x))) + t_0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.035], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 0.0305], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.035000000000000003

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 63.1%

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

   5. Applied rewrites 63.1%

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

   if -0.035000000000000003 < x < 0.030499999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.7%

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

   11. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-cos.f64 50.9%

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

   13. Applied rewrites 50.9%

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

   if 0.030499999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-cos.f64 63.1%

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, -0.0625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right), 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   8. Applied rewrites 63.1%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \color{blue}{-0.0625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + 0.6180339887498949 \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 80.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)\\ t_1 := \frac{2 + \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.035:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0305:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
           (* 0.38196601125010515 (cos y)))))
        (t_1
         (/
          (+
           2.0
           (*
            (* -0.0625 (* (pow (sin x) 2.0) (sqrt 2.0)))
            (- (cos x) (cos y))))
          t_0)))
   (if (<= x -0.035)
     t_1
     (if (<= x 0.0305)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (sqrt 2.0))
         (* (fma -0.0625 x (sin y)) (- (+ 1.0 (* -0.5 (pow x 2.0))) (cos y)))
         2.0)
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (0.38196601125010515 * cos(y)));
	double t_1 = (2.0 + ((-0.0625 * (pow(sin(x), 2.0) * sqrt(2.0))) * (cos(x) - cos(y)))) / t_0;
	double tmp;
	if (x <= -0.035) {
		tmp = t_1;
	} else if (x <= 0.0305) {
		tmp = fma((fma(-0.0625, sin(y), x) * sqrt(2.0)), (fma(-0.0625, x, sin(y)) * ((1.0 + (-0.5 * pow(x, 2.0))) - cos(y))), 2.0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(0.38196601125010515 * cos(y))))
	t_1 = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * sqrt(2.0))) * Float64(cos(x) - cos(y)))) / t_0)
	tmp = 0.0
	if (x <= -0.035)
		tmp = t_1;
	elseif (x <= 0.0305)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * sqrt(2.0)), Float64(fma(-0.0625, x, sin(y)) * Float64(Float64(1.0 + Float64(-0.5 * (x ^ 2.0))) - cos(y))), 2.0) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.035], t$95$1, If[LessEqual[x, 0.0305], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.035000000000000003 or 0.030499999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 63.1%

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

   5. Applied rewrites 63.1%

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

   if -0.035000000000000003 < x < 0.030499999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.7%

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

   11. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-cos.f64 50.9%

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

   13. Applied rewrites 50.9%

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

Alternative 20: 80.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sqrt{5} - 1}{2}\\ t_1 := 1 + -0.5 \cdot {x}^{2}\\ t_2 := 0.38196601125010515 \cdot \cos y\\ t_3 := \frac{2 + \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + t\_0 \cdot \cos x\right) + t\_2\right)}\\ \mathbf{if}\;x \leq -0.035:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.0305:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(t\_1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + t\_0 \cdot t\_1\right) + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_1 (+ 1.0 (* -0.5 (pow x 2.0))))
        (t_2 (* 0.38196601125010515 (cos y)))
        (t_3
         (/
          (+
           2.0
           (*
            (* -0.0625 (* (pow (sin x) 2.0) (sqrt 2.0)))
            (- (cos x) (cos y))))
          (* 3.0 (+ (+ 1.0 (* t_0 (cos x))) t_2)))))
   (if (<= x -0.035)
     t_3
     (if (<= x 0.0305)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (sqrt 2.0))
         (* (fma -0.0625 x (sin y)) (- t_1 (cos y)))
         2.0)
        (* 3.0 (+ (+ 1.0 (* t_0 t_1)) t_2)))
       t_3))))

C

double code(double x, double y) {
	double t_0 = (sqrt(5.0) - 1.0) / 2.0;
	double t_1 = 1.0 + (-0.5 * pow(x, 2.0));
	double t_2 = 0.38196601125010515 * cos(y);
	double t_3 = (2.0 + ((-0.0625 * (pow(sin(x), 2.0) * sqrt(2.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (t_0 * cos(x))) + t_2));
	double tmp;
	if (x <= -0.035) {
		tmp = t_3;
	} else if (x <= 0.0305) {
		tmp = fma((fma(-0.0625, sin(y), x) * sqrt(2.0)), (fma(-0.0625, x, sin(y)) * (t_1 - cos(y))), 2.0) / (3.0 * ((1.0 + (t_0 * t_1)) + t_2));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_1 = Float64(1.0 + Float64(-0.5 * (x ^ 2.0)))
	t_2 = Float64(0.38196601125010515 * cos(y))
	t_3 = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * sqrt(2.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_0 * cos(x))) + t_2)))
	tmp = 0.0
	if (x <= -0.035)
		tmp = t_3;
	elseif (x <= 0.0305)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * sqrt(2.0)), Float64(fma(-0.0625, x, sin(y)) * Float64(t_1 - cos(y))), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_0 * t_1)) + t_2)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.035], t$95$3, If[LessEqual[x, 0.0305], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.035000000000000003 or 0.030499999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 63.1%

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

   5. Applied rewrites 63.1%

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

   if -0.035000000000000003 < x < 0.030499999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.7%

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

   11. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 50.9%

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

   13. Applied rewrites 50.9%

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

   14. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 49.2%

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

   16. Applied rewrites 49.2%

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

Alternative 21: 80.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)\\ t_1 := \frac{2 + \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.035:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0305:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
           (* 0.38196601125010515 (cos y)))))
        (t_1
         (/
          (+
           2.0
           (*
            (* -0.0625 (* (pow (sin x) 2.0) (sqrt 2.0)))
            (- (cos x) (cos y))))
          t_0)))
   (if (<= x -0.035)
     t_1
     (if (<= x 0.0305)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (sqrt 2.0))
         (* (fma -0.0625 x (sin y)) (- 1.0 (cos y)))
         2.0)
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (0.38196601125010515 * cos(y)));
	double t_1 = (2.0 + ((-0.0625 * (pow(sin(x), 2.0) * sqrt(2.0))) * (cos(x) - cos(y)))) / t_0;
	double tmp;
	if (x <= -0.035) {
		tmp = t_1;
	} else if (x <= 0.0305) {
		tmp = fma((fma(-0.0625, sin(y), x) * sqrt(2.0)), (fma(-0.0625, x, sin(y)) * (1.0 - cos(y))), 2.0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(0.38196601125010515 * cos(y))))
	t_1 = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * sqrt(2.0))) * Float64(cos(x) - cos(y)))) / t_0)
	tmp = 0.0
	if (x <= -0.035)
		tmp = t_1;
	elseif (x <= 0.0305)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * sqrt(2.0)), Float64(fma(-0.0625, x, sin(y)) * Float64(1.0 - cos(y))), 2.0) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.035], t$95$1, If[LessEqual[x, 0.0305], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.035000000000000003 or 0.030499999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 63.1%

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

   5. Applied rewrites 63.1%

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

   if -0.035000000000000003 < x < 0.030499999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. lower-cos.f64 56.7%

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

   13. Applied rewrites 56.7%

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

Alternative 22: 80.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)\\ t_1 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.035:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0305:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
           (* 0.38196601125010515 (cos y)))))
        (t_1
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
          t_0)))
   (if (<= x -0.035)
     t_1
     (if (<= x 0.0305)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (sqrt 2.0))
         (* (fma -0.0625 x (sin y)) (- 1.0 (cos y)))
         2.0)
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (0.38196601125010515 * cos(y)));
	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 <= -0.035) {
		tmp = t_1;
	} else if (x <= 0.0305) {
		tmp = fma((fma(-0.0625, sin(y), x) * sqrt(2.0)), (fma(-0.0625, x, sin(y)) * (1.0 - cos(y))), 2.0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(0.38196601125010515 * 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))))) / t_0)
	tmp = 0.0
	if (x <= -0.035)
		tmp = t_1;
	elseif (x <= 0.0305)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * sqrt(2.0)), Float64(fma(-0.0625, x, sin(y)) * Float64(1.0 - cos(y))), 2.0) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $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] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.035], t$95$1, If[LessEqual[x, 0.0305], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.035000000000000003 or 0.030499999999999999 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 63.1%

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

   7. Applied rewrites 63.1%

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

   if -0.035000000000000003 < x < 0.030499999999999999

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. lower-cos.f64 56.7%

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

   13. Applied rewrites 56.7%

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

Alternative 23: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.96:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(1 + \mathsf{fma}\left(0.38196601125010515, \cos y, 0.5 \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

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)))))
          (*
           3.0
           (+
            (+ 1.0 (* (/ t_0 2.0) (cos x)))
            (* 0.38196601125010515 (cos y)))))))
   (if (<= x -0.00088)
     t_1
     (if (<= x 1.96)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (sqrt 2.0))
         (* (fma -0.0625 x (sin y)) (- (cos x) (cos y)))
         2.0)
        (* 3.0 (+ 1.0 (fma 0.38196601125010515 (cos y) (* 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))))) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + (0.38196601125010515 * cos(y))));
	double tmp;
	if (x <= -0.00088) {
		tmp = t_1;
	} else if (x <= 1.96) {
		tmp = fma((fma(-0.0625, sin(y), x) * sqrt(2.0)), (fma(-0.0625, x, sin(y)) * (cos(x) - cos(y))), 2.0) / (3.0 * (1.0 + fma(0.38196601125010515, cos(y), (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))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(0.38196601125010515 * cos(y)))))
	tmp = 0.0
	if (x <= -0.00088)
		tmp = t_1;
	elseif (x <= 1.96)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * sqrt(2.0)), Float64(fma(-0.0625, x, sin(y)) * Float64(cos(x) - cos(y))), 2.0) / Float64(3.0 * Float64(1.0 + fma(0.38196601125010515, cos(y), 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[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00088], t$95$1, If[LessEqual[x, 1.96], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.38196601125010515 * N[Cos[y], $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 < -8.8000000000000003e-4 or 1.96 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 63.1%

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

   7. Applied rewrites 63.1%

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

   if -8.8000000000000003e-4 < x < 1.96

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{6880887943736673}{18014398509481984} \cdot \cos y + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-sqrt.f64 50.5%

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

   13. Applied rewrites 50.5%

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

Alternative 24: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sqrt{5} - 1}{2}\\ t_1 := 0.38196601125010515 \cdot \cos y\\ t_2 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + t\_0 \cdot \cos x\right) + t\_1\right)}\\ \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + t\_0 \cdot 1\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_1 (* 0.38196601125010515 (cos y)))
        (t_2
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
          (* 3.0 (+ (+ 1.0 (* t_0 (cos x))) t_1)))))
   (if (<= x -0.00088)
     t_2
     (if (<= x 5.3)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (sqrt 2.0))
         (* (fma -0.0625 x (sin y)) (- 1.0 (cos y)))
         2.0)
        (* 3.0 (+ (+ 1.0 (* t_0 1.0)) t_1)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = (sqrt(5.0) - 1.0) / 2.0;
	double t_1 = 0.38196601125010515 * cos(y);
	double t_2 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 * ((1.0 + (t_0 * cos(x))) + t_1));
	double tmp;
	if (x <= -0.00088) {
		tmp = t_2;
	} else if (x <= 5.3) {
		tmp = fma((fma(-0.0625, sin(y), x) * sqrt(2.0)), (fma(-0.0625, x, sin(y)) * (1.0 - cos(y))), 2.0) / (3.0 * ((1.0 + (t_0 * 1.0)) + t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_1 = Float64(0.38196601125010515 * cos(y))
	t_2 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_0 * cos(x))) + t_1)))
	tmp = 0.0
	if (x <= -0.00088)
		tmp = t_2;
	elseif (x <= 5.3)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * sqrt(2.0)), Float64(fma(-0.0625, x, sin(y)) * Float64(1.0 - cos(y))), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_0 * 1.0)) + t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $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[(3.0 * N[(N[(1.0 + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00088], t$95$2, If[LessEqual[x, 5.3], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$0 * 1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.8000000000000003e-4 or 5.2999999999999998 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 63.1%

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

   7. Applied rewrites 63.1%

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

   if -8.8000000000000003e-4 < x < 5.2999999999999998

   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. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(\color{blue}{1} - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 56.7%

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

   14. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{1}\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   15. Step-by-step derivation

   16. Applied rewrites 54.5%

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

Alternative 25: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x \cdot t\_1\\ \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t\_0}{1 + \mathsf{fma}\left(0.5, t\_2, 0.5 \cdot 0.7639320225002103\right)}\\ \mathbf{elif}\;x \leq 5.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot 1\right) + 0.38196601125010515 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{3 + 1.5 \cdot \left(\left(3 + t\_2\right) - \sqrt{5}\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (cos x) t_1)))
   (if (<= x -0.00088)
     (*
      0.3333333333333333
      (/ t_0 (+ 1.0 (fma 0.5 t_2 (* 0.5 0.7639320225002103)))))
     (if (<= x 5.3)
       (/
        (fma
         (* (fma -0.0625 (sin y) x) (sqrt 2.0))
         (* (fma -0.0625 x (sin y)) (- 1.0 (cos y)))
         2.0)
        (*
         3.0
         (+ (+ 1.0 (* (/ t_1 2.0) 1.0)) (* 0.38196601125010515 (cos y)))))
       (/ t_0 (+ 3.0 (* 1.5 (- (+ 3.0 t_2) (sqrt 5.0)))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) * t_1;
	double tmp;
	if (x <= -0.00088) {
		tmp = 0.3333333333333333 * (t_0 / (1.0 + fma(0.5, t_2, (0.5 * 0.7639320225002103))));
	} else if (x <= 5.3) {
		tmp = fma((fma(-0.0625, sin(y), x) * sqrt(2.0)), (fma(-0.0625, x, sin(y)) * (1.0 - cos(y))), 2.0) / (3.0 * ((1.0 + ((t_1 / 2.0) * 1.0)) + (0.38196601125010515 * cos(y))));
	} else {
		tmp = t_0 / (3.0 + (1.5 * ((3.0 + t_2) - sqrt(5.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) * t_1)
	tmp = 0.0
	if (x <= -0.00088)
		tmp = Float64(0.3333333333333333 * Float64(t_0 / Float64(1.0 + fma(0.5, t_2, Float64(0.5 * 0.7639320225002103)))));
	elseif (x <= 5.3)
		tmp = Float64(fma(Float64(fma(-0.0625, sin(y), x) * sqrt(2.0)), Float64(fma(-0.0625, x, sin(y)) * Float64(1.0 - cos(y))), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * 1.0)) + Float64(0.38196601125010515 * cos(y)))));
	else
		tmp = Float64(t_0 / Float64(3.0 + Float64(1.5 * Float64(Float64(3.0 + t_2) - sqrt(5.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[x, -0.00088], N[(0.3333333333333333 * N[(t$95$0 / N[(1.0 + N[(0.5 * t$95$2 + N[(0.5 * 0.7639320225002103), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.3], N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 + N[(1.5 * N[(N[(3.0 + t$95$2), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.8000000000000003e-4

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.8%

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

   5. Evaluated real constant 60.8%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot 0.7639320225002103\right)} \]

   if -8.8000000000000003e-4 < x < 5.2999999999999998

   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. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \color{blue}{x}\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{x}, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(\color{blue}{1} - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 56.7%

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

   14. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{1}\right) + 0.38196601125010515 \cdot \cos y\right)} \]
   15. Step-by-step derivation

   16. Applied rewrites 54.5%

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

   if 5.2999999999999998 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 60.8%

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

Alternative 26: 78.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x \cdot t\_1\\ \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t\_0}{1 + \mathsf{fma}\left(0.5, t\_2, 0.5 \cdot 0.7639320225002103\right)}\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{3 + 1.5 \cdot \left(\left(3 + t\_2\right) - \sqrt{5}\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (cos x) t_1)))
   (if (<= x -0.00088)
     (*
      0.3333333333333333
      (/ t_0 (+ 1.0 (fma 0.5 t_2 (* 0.5 0.7639320225002103)))))
     (if (<= x 200.0)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (* 3.0 (+ 1.0 (fma 0.5 (* (cos y) (- 3.0 (sqrt 5.0))) (* 0.5 t_1)))))
       (/ t_0 (+ 3.0 (* 1.5 (- (+ 3.0 t_2) (sqrt 5.0)))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) * t_1;
	double tmp;
	if (x <= -0.00088) {
		tmp = 0.3333333333333333 * (t_0 / (1.0 + fma(0.5, t_2, (0.5 * 0.7639320225002103))));
	} else if (x <= 200.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (1.0 + fma(0.5, (cos(y) * (3.0 - sqrt(5.0))), (0.5 * t_1))));
	} else {
		tmp = t_0 / (3.0 + (1.5 * ((3.0 + t_2) - sqrt(5.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) * t_1)
	tmp = 0.0
	if (x <= -0.00088)
		tmp = Float64(0.3333333333333333 * Float64(t_0 / Float64(1.0 + fma(0.5, t_2, Float64(0.5 * 0.7639320225002103)))));
	elseif (x <= 200.0)
		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(1.0 + fma(0.5, Float64(cos(y) * Float64(3.0 - sqrt(5.0))), Float64(0.5 * t_1)))));
	else
		tmp = Float64(t_0 / Float64(3.0 + Float64(1.5 * Float64(Float64(3.0 + t_2) - sqrt(5.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[x, -0.00088], N[(0.3333333333333333 * N[(t$95$0 / N[(1.0 + N[(0.5 * t$95$2 + N[(0.5 * 0.7639320225002103), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 200.0], 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[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 + N[(1.5 * N[(N[(3.0 + t$95$2), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.8000000000000003e-4

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.8%

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

   5. Evaluated real constant 60.8%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot 0.7639320225002103\right)} \]

   if -8.8000000000000003e-4 < x < 200

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 59.5%

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 59.0%

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

   7. Applied rewrites 59.0%

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

   if 200 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 60.8%

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

Alternative 27: 78.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} - 1\right)\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\\ \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t\_1}{1 + \mathsf{fma}\left(0.5, t\_0, 0.5 \cdot 0.7639320225002103\right)}\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 + 1.5 \cdot \left(\left(3 + t\_0\right) - \sqrt{5}\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (- (sqrt 5.0) 1.0)))
        (t_1
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))))
   (if (<= x -0.00088)
     (*
      0.3333333333333333
      (/ t_1 (+ 1.0 (fma 0.5 t_0 (* 0.5 0.7639320225002103)))))
     (if (<= x 200.0)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* (cos y) (- 3.0 (sqrt 5.0)))) 1.0))))
       (/ t_1 (+ 3.0 (* 1.5 (- (+ 3.0 t_0) (sqrt 5.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sqrt(5.0) - 1.0);
	double t_1 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))));
	double tmp;
	if (x <= -0.00088) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + fma(0.5, t_0, (0.5 * 0.7639320225002103))));
	} else if (x <= 200.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + (cos(y) * (3.0 - sqrt(5.0)))) - 1.0)));
	} else {
		tmp = t_1 / (3.0 + (1.5 * ((3.0 + t_0) - sqrt(5.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sqrt(5.0) - 1.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))
	tmp = 0.0
	if (x <= -0.00088)
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + fma(0.5, t_0, Float64(0.5 * 0.7639320225002103)))));
	elseif (x <= 200.0)
		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(cos(y) * Float64(3.0 - sqrt(5.0)))) - 1.0))));
	else
		tmp = Float64(t_1 / Float64(3.0 + Float64(1.5 * Float64(Float64(3.0 + t_0) - sqrt(5.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00088], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(0.5 * t$95$0 + N[(0.5 * 0.7639320225002103), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 200.0], 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[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 + N[(1.5 * N[(N[(3.0 + t$95$0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.8000000000000003e-4

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.8%

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

   5. Evaluated real constant 60.8%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot 0.7639320225002103\right)} \]

   if -8.8000000000000003e-4 < x < 200

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 59.0%

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

   if 200 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 60.8%

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

Alternative 28: 78.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}\\ \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
          (+
           3.0
           (* 1.5 (- (+ 3.0 (* (cos x) (- (sqrt 5.0) 1.0))) (sqrt 5.0)))))))
   (if (<= x -0.00088)
     t_0
     (if (<= x 200.0)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* (cos y) (- 3.0 (sqrt 5.0)))) 1.0))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (1.5 * ((3.0 + (cos(x) * (sqrt(5.0) - 1.0))) - sqrt(5.0))));
	double tmp;
	if (x <= -0.00088) {
		tmp = t_0;
	} else if (x <= 200.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + (cos(y) * (3.0 - sqrt(5.0)))) - 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (3.0d0 + (1.5d0 * ((3.0d0 + (cos(x) * (sqrt(5.0d0) - 1.0d0))) - sqrt(5.0d0))))
    if (x <= (-0.00088d0)) then
        tmp = t_0
    else if (x <= 200.0d0) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * ((sqrt(5.0d0) + (cos(y) * (3.0d0 - sqrt(5.0d0)))) - 1.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (3.0 + (1.5 * ((3.0 + (Math.cos(x) * (Math.sqrt(5.0) - 1.0))) - Math.sqrt(5.0))));
	double tmp;
	if (x <= -0.00088) {
		tmp = t_0;
	} else if (x <= 200.0) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (1.5 * ((Math.sqrt(5.0) + (Math.cos(y) * (3.0 - Math.sqrt(5.0)))) - 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (3.0 + (1.5 * ((3.0 + (math.cos(x) * (math.sqrt(5.0) - 1.0))) - math.sqrt(5.0))))
	tmp = 0
	if x <= -0.00088:
		tmp = t_0
	elif x <= 200.0:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (1.5 * ((math.sqrt(5.0) + (math.cos(y) * (3.0 - math.sqrt(5.0)))) - 1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(3.0 + Float64(cos(x) * Float64(sqrt(5.0) - 1.0))) - sqrt(5.0)))))
	tmp = 0.0
	if (x <= -0.00088)
		tmp = t_0;
	elseif (x <= 200.0)
		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(cos(y) * Float64(3.0 - sqrt(5.0)))) - 1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (1.5 * ((3.0 + (cos(x) * (sqrt(5.0) - 1.0))) - sqrt(5.0))));
	tmp = 0.0;
	if (x <= -0.00088)
		tmp = t_0;
	elseif (x <= 200.0)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((sqrt(5.0) + (cos(y) * (3.0 - sqrt(5.0)))) - 1.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00088], t$95$0, If[LessEqual[x, 200.0], 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[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.8000000000000003e-4 or 200 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 60.8%

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

   if -8.8000000000000003e-4 < x < 200

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 59.0%

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

Alternative 29: 78.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}\\ \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.618033988749895 + 0.38196601125010515 \cdot \cos y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
          (+
           3.0
           (* 1.5 (- (+ 3.0 (* (cos x) (- (sqrt 5.0) 1.0))) (sqrt 5.0)))))))
   (if (<= x -0.00088)
     t_0
     (if (<= x 200.0)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
         (+ 1.618033988749895 (* 0.38196601125010515 (cos y)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (1.5 * ((3.0 + (cos(x) * (sqrt(5.0) - 1.0))) - sqrt(5.0))));
	double tmp;
	if (x <= -0.00088) {
		tmp = t_0;
	} else if (x <= 200.0) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (1.618033988749895 + (0.38196601125010515 * cos(y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (3.0d0 + (1.5d0 * ((3.0d0 + (cos(x) * (sqrt(5.0d0) - 1.0d0))) - sqrt(5.0d0))))
    if (x <= (-0.00088d0)) then
        tmp = t_0
    else if (x <= 200.0d0) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (1.618033988749895d0 + (0.38196601125010515d0 * cos(y))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (3.0 + (1.5 * ((3.0 + (Math.cos(x) * (Math.sqrt(5.0) - 1.0))) - Math.sqrt(5.0))));
	double tmp;
	if (x <= -0.00088) {
		tmp = t_0;
	} else if (x <= 200.0) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (1.618033988749895 + (0.38196601125010515 * Math.cos(y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (3.0 + (1.5 * ((3.0 + (math.cos(x) * (math.sqrt(5.0) - 1.0))) - math.sqrt(5.0))))
	tmp = 0
	if x <= -0.00088:
		tmp = t_0
	elif x <= 200.0:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (1.618033988749895 + (0.38196601125010515 * math.cos(y))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(3.0 + Float64(cos(x) * Float64(sqrt(5.0) - 1.0))) - sqrt(5.0)))))
	tmp = 0.0
	if (x <= -0.00088)
		tmp = t_0;
	elseif (x <= 200.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(1.618033988749895 + Float64(0.38196601125010515 * cos(y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (1.5 * ((3.0 + (cos(x) * (sqrt(5.0) - 1.0))) - sqrt(5.0))));
	tmp = 0.0;
	if (x <= -0.00088)
		tmp = t_0;
	elseif (x <= 200.0)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (1.618033988749895 + (0.38196601125010515 * cos(y))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00088], t$95$0, If[LessEqual[x, 200.0], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.618033988749895 + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.8000000000000003e-4 or 200 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 60.8%

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

   if -8.8000000000000003e-4 < x < 200

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]

   8. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.618033988749895 + 0.38196601125010515 \cdot \cos y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 78.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - 1\\ \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{1.381966011250105 + 0.6180339887498949 \cdot \cos x}\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.618033988749895 + 0.38196601125010515 \cdot \cos y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625 \cdot \left(t\_0 \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0)))
   (if (<= x -0.00088)
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_0))))
       (+ 1.381966011250105 (* 0.6180339887498949 (cos x)))))
     (if (<= x 200.0)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
         (+ 1.618033988749895 (* 0.38196601125010515 (cos y)))))
       (/
        (*
         (fma
          (* 0.0625 (* t_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) {
	double t_0 = cos(x) - 1.0;
	double tmp;
	if (x <= -0.00088) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_0)))) / (1.381966011250105 + (0.6180339887498949 * cos(x))));
	} else if (x <= 200.0) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (1.618033988749895 + (0.38196601125010515 * cos(y))));
	} else {
		tmp = (fma((0.0625 * (t_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);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	tmp = 0.0
	if (x <= -0.00088)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_0)))) / Float64(1.381966011250105 + Float64(0.6180339887498949 * cos(x)))));
	elseif (x <= 200.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(1.618033988749895 + Float64(0.38196601125010515 * cos(y)))));
	else
		tmp = Float64(Float64(fma(Float64(0.0625 * Float64(t_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
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.00088], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.381966011250105 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 200.0], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.618033988749895 + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[(t$95$0 * 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. Split input into 3 regimes

2. if x < -8.8000000000000003e-4

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]

   8. Applied rewrites 60.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1.381966011250105 + 0.6180339887498949 \cdot \cos x}} \]

   if -8.8000000000000003e-4 < x < 200

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]

   8. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.618033988749895 + 0.38196601125010515 \cdot \cos y}} \]

   if 200 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 60.8%

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

Alternative 31: 78.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \cos x - 1\\ \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{1.381966011250105 + 0.6180339887498949 \cdot \cos x}\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.618033988749895 + 0.38196601125010515 \cdot \cos y}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0625 \cdot \left(t\_0 \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right) \cdot 0.3333333333333333\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0)))
   (if (<= x -0.00088)
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_0))))
       (+ 1.381966011250105 (* 0.6180339887498949 (cos x)))))
     (if (<= x 200.0)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
         (+ 1.618033988749895 (* 0.38196601125010515 (cos y)))))
       (*
        (*
         (fma
          (* -0.0625 (* t_0 (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x))))
          2.0)
         0.3333333333333333)
        (/
         1.0
         (fma
          (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0)))
          0.5
          1.0)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double tmp;
	if (x <= -0.00088) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_0)))) / (1.381966011250105 + (0.6180339887498949 * cos(x))));
	} else if (x <= 200.0) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (1.618033988749895 + (0.38196601125010515 * cos(y))));
	} else {
		tmp = (fma((-0.0625 * (t_0 * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) * 0.3333333333333333) * (1.0 / fma(fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	tmp = 0.0
	if (x <= -0.00088)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_0)))) / Float64(1.381966011250105 + Float64(0.6180339887498949 * cos(x)))));
	elseif (x <= 200.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(1.618033988749895 + Float64(0.38196601125010515 * cos(y)))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(t_0 * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0) * 0.3333333333333333) * Float64(1.0 / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.00088], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.381966011250105 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 200.0], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.618033988749895 + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(t$95$0 * 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[(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]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.8000000000000003e-4

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]

   8. Applied rewrites 60.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1.381966011250105 + 0.6180339887498949 \cdot \cos x}} \]

   if -8.8000000000000003e-4 < x < 200

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]

   8. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.618033988749895 + 0.38196601125010515 \cdot \cos y}} \]

   if 200 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 60.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 60.8%

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

Alternative 32: 78.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1.381966011250105 + 0.6180339887498949 \cdot \cos x}\\ \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 200:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.618033988749895 + 0.38196601125010515 \cdot \cos y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          0.3333333333333333
          (/
           (+
            2.0
            (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
           (+ 1.381966011250105 (* 0.6180339887498949 (cos x)))))))
   (if (<= x -0.00088)
     t_0
     (if (<= x 200.0)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
         (+ 1.618033988749895 (* 0.38196601125010515 (cos y)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.381966011250105 + (0.6180339887498949 * cos(x))));
	double tmp;
	if (x <= -0.00088) {
		tmp = t_0;
	} else if (x <= 200.0) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (1.618033988749895 + (0.38196601125010515 * cos(y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (1.381966011250105d0 + (0.6180339887498949d0 * cos(x))))
    if (x <= (-0.00088d0)) then
        tmp = t_0
    else if (x <= 200.0d0) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (1.618033988749895d0 + (0.38196601125010515d0 * cos(y))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (1.381966011250105 + (0.6180339887498949 * Math.cos(x))));
	double tmp;
	if (x <= -0.00088) {
		tmp = t_0;
	} else if (x <= 200.0) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (1.618033988749895 + (0.38196601125010515 * Math.cos(y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (1.381966011250105 + (0.6180339887498949 * math.cos(x))))
	tmp = 0
	if x <= -0.00088:
		tmp = t_0
	elif x <= 200.0:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (1.618033988749895 + (0.38196601125010515 * math.cos(y))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(1.381966011250105 + Float64(0.6180339887498949 * cos(x)))))
	tmp = 0.0
	if (x <= -0.00088)
		tmp = t_0;
	elseif (x <= 200.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(1.618033988749895 + Float64(0.38196601125010515 * cos(y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.381966011250105 + (0.6180339887498949 * cos(x))));
	tmp = 0.0;
	if (x <= -0.00088)
		tmp = t_0;
	elseif (x <= 200.0)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (1.618033988749895 + (0.38196601125010515 * cos(y))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.381966011250105 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00088], t$95$0, If[LessEqual[x, 200.0], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.618033988749895 + N[(0.38196601125010515 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.8000000000000003e-4 or 200 < x

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]

   8. Applied rewrites 60.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1.381966011250105 + 0.6180339887498949 \cdot \cos x}} \]

   if -8.8000000000000003e-4 < x < 200

   1. Initial program 99.3%

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

   2. Evaluated real constant 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Evaluated real constant 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{910872158600853}{562949953421312} + \frac{6880887943736673}{18014398509481984} \cdot \cos y}} \]

   8. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1.618033988749895 + 0.38196601125010515 \cdot \cos y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 33: 60.9% accurate, 2.3× speedup

Math

\[0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1.381966011250105 + 0.6180339887498949 \cdot \cos x} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
   (+ 1.381966011250105 (* 0.6180339887498949 (cos x))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.381966011250105 + (0.6180339887498949 * cos(x))));
}

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 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (1.381966011250105d0 + (0.6180339887498949d0 * cos(x))))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (1.381966011250105 + (0.6180339887498949 * Math.cos(x))));
}

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (1.381966011250105 + (0.6180339887498949 * math.cos(x))))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(1.381966011250105 + Float64(0.6180339887498949 * cos(x)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.381966011250105 + (0.6180339887498949 * cos(x))));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.381966011250105 + N[(0.6180339887498949 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Evaluated real constant 99.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Evaluated real constant 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \color{blue}{0.6180339887498949} \cdot \cos x\right) + 0.38196601125010515 \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\frac{24895286453218657}{18014398509481984} + \frac{347922205179541}{562949953421312} \cdot \cos x}} \]

8. Applied rewrites 60.9%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1.381966011250105 + 0.6180339887498949 \cdot \cos x}} \]
9. Add Preprocessing

Alternative 34: 43.2% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 60.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 43.2%

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

Alternative 35: 40.5% accurate, 316.7× speedup

Math

\[0.3333333333333333 \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 60.8%

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

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 40.5%

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

7. Applied rewrites 40.5%

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

8. Evaluated real constant 40.5%

   \[\leadsto 0.3333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025187 
(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))))))