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

Percentage Accurate: 99.3% → 99.3%

Time: 9.7s

Alternatives: 31

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-cos.f64 99.3

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

5. Applied rewrites 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. associate-+r+ N/A

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

5. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.2%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 99.2%

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

Alternative 7: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y\\ t_3 := \sqrt{5} - 1\\ t_4 := \frac{\frac{\mathsf{fma}\left(t\_0, \sin y \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right), 2\right)}{3}}{\mathsf{fma}\left(0.5 \cdot t\_1, \cos y, \mathsf{fma}\left(t\_3 \cdot \cos x, 0.5, 1\right)\right)}\\ \mathbf{if}\;y \leq -0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;y \leq 0.115:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{t\_2}{16}\right)\right) \cdot \left(t\_2 - \frac{\sin x}{16}\right)\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{t\_3}{2} \cdot \cos x\right) + \frac{t\_1}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* (fma (* y y) -0.16666666666666666 1.0) y))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4
         (/
          (/
           (fma
            t_0
            (* (sin y) (* (- (sin x) (* (sin y) 0.0625)) (sqrt 2.0)))
            2.0)
           3.0)
          (fma (* 0.5 t_1) (cos y) (fma (* t_3 (cos x)) 0.5 1.0)))))
   (if (<= y -0.2)
     t_4
     (if (<= y 0.115)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ t_2 16.0))) (- t_2 (/ (sin x) 16.0)))
          t_0))
        (* 3.0 (+ (+ 1.0 (* (/ t_3 2.0) (cos x))) (* (/ t_1 2.0) (cos y)))))
       t_4))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((y * y), -0.16666666666666666, 1.0) * y;
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = (fma(t_0, (sin(y) * ((sin(x) - (sin(y) * 0.0625)) * sqrt(2.0))), 2.0) / 3.0) / fma((0.5 * t_1), cos(y), fma((t_3 * cos(x)), 0.5, 1.0));
	double tmp;
	if (y <= -0.2) {
		tmp = t_4;
	} else if (y <= 0.115) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (t_2 / 16.0))) * (t_2 - (sin(x) / 16.0))) * t_0)) / (3.0 * ((1.0 + ((t_3 / 2.0) * cos(x))) + ((t_1 / 2.0) * cos(y))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y)
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = Float64(Float64(fma(t_0, Float64(sin(y) * Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * sqrt(2.0))), 2.0) / 3.0) / fma(Float64(0.5 * t_1), cos(y), fma(Float64(t_3 * cos(x)), 0.5, 1.0)))
	tmp = 0.0
	if (y <= -0.2)
		tmp = t_4;
	elseif (y <= 0.115)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(t_2 / 16.0))) * Float64(t_2 - Float64(sin(x) / 16.0))) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_3 / 2.0) * cos(x))) + Float64(Float64(t_1 / 2.0) * cos(y)))));
	else
		tmp = t_4;
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.2], t$95$4, If[LessEqual[y, 0.115], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(t$95$2 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.20000000000000001 or 0.115000000000000005 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-cos.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lift-sin.f64 64.3

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

   8. Applied rewrites 64.3%

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

   if -0.20000000000000001 < y < 0.115000000000000005

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.0

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 49.5

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

   7. Applied rewrites 49.5%

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

Alternative 8: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{\mathsf{fma}\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \cos x - \cos y, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_1 \cdot \cos y\right), 1\right)}\\ t_4 := \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\\ \mathbf{if}\;x \leq -0.145:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_4 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_4}{16}\right)\right) \cdot \left(t\_0 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot t\_0\right) + \frac{t\_1}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma 0.041666666666666664 (* x x) -0.5) (* x x) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3
         (/
          (*
           (fma
            (* (* (sin x) (sqrt 2.0)) (- (sin y) (* (sin x) 0.0625)))
            (- (cos x) (cos y))
            2.0)
           0.3333333333333333)
          (fma 0.5 (fma t_2 (cos x) (* t_1 (cos y))) 1.0)))
        (t_4
         (*
          (fma
           (fma 0.008333333333333333 (* x x) -0.16666666666666666)
           (* x x)
           1.0)
          x)))
   (if (<= x -0.145)
     t_3
     (if (<= x 1.9e-5)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_4 (/ (sin y) 16.0))) (- (sin y) (/ t_4 16.0)))
          (- t_0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) t_0)) (* (/ t_1 2.0) (cos y)))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.041666666666666664, (x * x), -0.5), (x * x), 1.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = (fma(((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) * 0.0625))), (cos(x) - cos(y)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_2, cos(x), (t_1 * cos(y))), 1.0);
	double t_4 = fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * x;
	double tmp;
	if (x <= -0.145) {
		tmp = t_3;
	} else if (x <= 1.9e-5) {
		tmp = (2.0 + (((sqrt(2.0) * (t_4 - (sin(y) / 16.0))) * (sin(y) - (t_4 / 16.0))) * (t_0 - cos(y)))) / (3.0 * ((1.0 + ((t_2 / 2.0) * t_0)) + ((t_1 / 2.0) * cos(y))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(0.041666666666666664, Float64(x * x), -0.5), Float64(x * x), 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(Float64(fma(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) * 0.0625))), Float64(cos(x) - cos(y)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_2, cos(x), Float64(t_1 * cos(y))), 1.0))
	t_4 = Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x)
	tmp = 0.0
	if (x <= -0.145)
		tmp = t_3;
	elseif (x <= 1.9e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_4 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_4 / 16.0))) * Float64(t_0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * t_0)) + Float64(Float64(t_1 / 2.0) * cos(y)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -0.145], t$95$3, If[LessEqual[x, 1.9e-5], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$4 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$4 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.14499999999999999 or 1.9000000000000001e-5 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sqrt.f64 63.5

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

   8. Applied rewrites 63.5%

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

   if -0.14499999999999999 < x < 1.9000000000000001e-5

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. sub-flip N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{x}}^{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}, {x}^{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right), {\color{blue}{x}}^{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot \color{blue}{x}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 50.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot \color{blue}{x}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 50.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. sub-flip N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {\color{blue}{x}}^{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}, {x}^{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right), {\color{blue}{x}}^{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), {x}^{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot \color{blue}{x}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 50.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot \color{blue}{x}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 50.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2} + 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-flip N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {x}^{2}, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}, {x}^{2}, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 50.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 50.3%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2} + 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. sub-flip N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {x}^{2}, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}, {x}^{2}, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       11. pow2 N/A

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       12. lift-*.f64 50.0

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 50.0%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 79.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := -0.0625 \cdot {\sin y}^{2}\\ t_3 := y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ t_4 := \sqrt{5} - 1\\ t_5 := \mathsf{fma}\left(0.5 \cdot t\_1, \cos y, \mathsf{fma}\left(0.5 \cdot t\_4, \cos x, 1\right)\right)\\ \mathbf{if}\;y \leq -0.2:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3}}{t\_5}\\ \mathbf{elif}\;y \leq 0.115:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\sin x - t\_3 \cdot 0.0625\right) \cdot \sqrt{2}\right) \cdot \left(t\_3 - \sin x \cdot 0.0625\right), t\_0, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_4, \cos x, t\_1 \cdot \cos y\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0, t\_2 \cdot \sqrt{2}, 2\right)}{3}}{t\_5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* -0.0625 (pow (sin y) 2.0)))
        (t_3 (* y (- 1.0 (* 0.16666666666666666 (* y y)))))
        (t_4 (- (sqrt 5.0) 1.0))
        (t_5 (fma (* 0.5 t_1) (cos y) (fma (* 0.5 t_4) (cos x) 1.0))))
   (if (<= y -0.2)
     (/ (/ (fma t_2 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0) 3.0) t_5)
     (if (<= y 0.115)
       (/
        (*
         (fma
          (*
           (* (- (sin x) (* t_3 0.0625)) (sqrt 2.0))
           (- t_3 (* (sin x) 0.0625)))
          t_0
          2.0)
         0.3333333333333333)
        (fma 0.5 (fma t_4 (cos x) (* t_1 (cos y))) 1.0))
       (/ (/ (fma t_0 (* t_2 (sqrt 2.0)) 2.0) 3.0) t_5)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = -0.0625 * pow(sin(y), 2.0);
	double t_3 = y * (1.0 - (0.16666666666666666 * (y * y)));
	double t_4 = sqrt(5.0) - 1.0;
	double t_5 = fma((0.5 * t_1), cos(y), fma((0.5 * t_4), cos(x), 1.0));
	double tmp;
	if (y <= -0.2) {
		tmp = (fma(t_2, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / t_5;
	} else if (y <= 0.115) {
		tmp = (fma((((sin(x) - (t_3 * 0.0625)) * sqrt(2.0)) * (t_3 - (sin(x) * 0.0625))), t_0, 2.0) * 0.3333333333333333) / fma(0.5, fma(t_4, cos(x), (t_1 * cos(y))), 1.0);
	} else {
		tmp = (fma(t_0, (t_2 * sqrt(2.0)), 2.0) / 3.0) / t_5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(-0.0625 * (sin(y) ^ 2.0))
	t_3 = Float64(y * Float64(1.0 - Float64(0.16666666666666666 * Float64(y * y))))
	t_4 = Float64(sqrt(5.0) - 1.0)
	t_5 = fma(Float64(0.5 * t_1), cos(y), fma(Float64(0.5 * t_4), cos(x), 1.0))
	tmp = 0.0
	if (y <= -0.2)
		tmp = Float64(Float64(fma(t_2, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / t_5);
	elseif (y <= 0.115)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(sin(x) - Float64(t_3 * 0.0625)) * sqrt(2.0)) * Float64(t_3 - Float64(sin(x) * 0.0625))), t_0, 2.0) * 0.3333333333333333) / fma(0.5, fma(t_4, cos(x), Float64(t_1 * cos(y))), 1.0));
	else
		tmp = Float64(Float64(fma(t_0, Float64(t_2 * sqrt(2.0)), 2.0) / 3.0) / t_5);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(1.0 - N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * t$95$4), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.2], N[(N[(N[(t$95$2 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[y, 0.115], N[(N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(t$95$3 * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(t$95$4 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / t$95$5), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.20000000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 62.6

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

   6. Applied rewrites 62.6%

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

   if -0.20000000000000001 < y < 0.115000000000000005

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.1

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

   8. Applied rewrites 50.1%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. lower--.f64 N/A

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

       4. metadata-eval N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 49.6

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

   11. Applied rewrites 49.6%

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

   if 0.115000000000000005 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sqrt.f64 62.6

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

   6. Applied rewrites 62.6%

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

Alternative 10: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := -0.0625 \cdot {\sin y}^{2}\\ t_2 := \sqrt{5} - 1\\ t_3 := \mathsf{fma}\left(0.5 \cdot t\_0, \cos y, \mathsf{fma}\left(0.5 \cdot t\_2, \cos x, 1\right)\right)\\ t_4 := 1 - 0.5 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq -30500:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3}}{t\_3}\\ \mathbf{elif}\;y \leq 0.015:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \cos x - t\_4, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot t\_4\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, t\_1 \cdot \sqrt{2}, 2\right)}{3}}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* -0.0625 (pow (sin y) 2.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (fma (* 0.5 t_0) (cos y) (fma (* 0.5 t_2) (cos x) 1.0)))
        (t_4 (- 1.0 (* 0.5 (* y y)))))
   (if (<= y -30500.0)
     (/ (/ (fma t_1 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0) 3.0) t_3)
     (if (<= y 0.015)
       (/
        (*
         (fma
          (*
           (* (- (sin x) (* (sin y) 0.0625)) (sqrt 2.0))
           (- (sin y) (* (sin x) 0.0625)))
          (- (cos x) t_4)
          2.0)
         0.3333333333333333)
        (fma 0.5 (fma t_2 (cos x) (* t_0 t_4)) 1.0))
       (/ (/ (fma (- (cos x) (cos y)) (* t_1 (sqrt 2.0)) 2.0) 3.0) t_3)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = -0.0625 * pow(sin(y), 2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma((0.5 * t_0), cos(y), fma((0.5 * t_2), cos(x), 1.0));
	double t_4 = 1.0 - (0.5 * (y * y));
	double tmp;
	if (y <= -30500.0) {
		tmp = (fma(t_1, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / t_3;
	} else if (y <= 0.015) {
		tmp = (fma((((sin(x) - (sin(y) * 0.0625)) * sqrt(2.0)) * (sin(y) - (sin(x) * 0.0625))), (cos(x) - t_4), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_2, cos(x), (t_0 * t_4)), 1.0);
	} else {
		tmp = (fma((cos(x) - cos(y)), (t_1 * sqrt(2.0)), 2.0) / 3.0) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(-0.0625 * (sin(y) ^ 2.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = fma(Float64(0.5 * t_0), cos(y), fma(Float64(0.5 * t_2), cos(x), 1.0))
	t_4 = Float64(1.0 - Float64(0.5 * Float64(y * y)))
	tmp = 0.0
	if (y <= -30500.0)
		tmp = Float64(Float64(fma(t_1, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / t_3);
	elseif (y <= 0.015)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) * 0.0625))), Float64(cos(x) - t_4), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_2, cos(x), Float64(t_0 * t_4)), 1.0));
	else
		tmp = Float64(Float64(fma(Float64(cos(x) - cos(y)), Float64(t_1 * sqrt(2.0)), 2.0) / 3.0) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -30500.0], N[(N[(N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 0.015], N[(N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - t$95$4), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * t$95$4), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -30500

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 62.6

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

   6. Applied rewrites 62.6%

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

   if -30500 < y < 0.014999999999999999

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 49.6

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

   8. Applied rewrites 49.6%

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

   9. Taylor expanded in y around 0

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

       1. fp-cancel-sign-sub-inv N/A

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

       2. metadata-eval N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 52.2

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

   11. Applied rewrites 52.2%

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

   if 0.014999999999999999 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sqrt.f64 62.6

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

   6. Applied rewrites 62.6%

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

Alternative 11: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := -0.0625 \cdot {\sin y}^{2}\\ t_2 := \sqrt{5} - 1\\ t_3 := \mathsf{fma}\left(0.5 \cdot t\_0, \cos y, \mathsf{fma}\left(0.5 \cdot t\_2, \cos x, 1\right)\right)\\ \mathbf{if}\;y \leq -0.021:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3}}{t\_3}\\ \mathbf{elif}\;y \leq 0.021:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot \left(y \cdot y\right)\right)\right)}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos x, t\_2, t\_0\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.75, t\_0, 0.0625 \cdot \left(\left(y \cdot y\right) \cdot t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\cos x - \cos y, t\_1 \cdot \sqrt{2}, 2\right)}{3}}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* -0.0625 (pow (sin y) 2.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (fma (* 0.5 t_0) (cos y) (fma (* 0.5 t_2) (cos x) 1.0))))
   (if (<= y -0.021)
     (/ (/ (fma t_1 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0) 3.0) t_3)
     (if (<= y 0.021)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          (- (cos x) (+ 1.0 (* -0.5 (* y y))))))
        (fma
         3.0
         (+ 1.0 (* 0.5 (fma (cos x) t_2 t_0)))
         (* (* y y) (fma -0.75 t_0 (* 0.0625 (* (* y y) t_0))))))
       (/ (/ (fma (- (cos x) (cos y)) (* t_1 (sqrt 2.0)) 2.0) 3.0) t_3)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = -0.0625 * pow(sin(y), 2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma((0.5 * t_0), cos(y), fma((0.5 * t_2), cos(x), 1.0));
	double tmp;
	if (y <= -0.021) {
		tmp = (fma(t_1, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / t_3;
	} else if (y <= 0.021) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * (cos(x) - (1.0 + (-0.5 * (y * y)))))) / fma(3.0, (1.0 + (0.5 * fma(cos(x), t_2, t_0))), ((y * y) * fma(-0.75, t_0, (0.0625 * ((y * y) * t_0)))));
	} else {
		tmp = (fma((cos(x) - cos(y)), (t_1 * sqrt(2.0)), 2.0) / 3.0) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(-0.0625 * (sin(y) ^ 2.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = fma(Float64(0.5 * t_0), cos(y), fma(Float64(0.5 * t_2), cos(x), 1.0))
	tmp = 0.0
	if (y <= -0.021)
		tmp = Float64(Float64(fma(t_1, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / t_3);
	elseif (y <= 0.021)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * Float64(cos(x) - Float64(1.0 + Float64(-0.5 * Float64(y * y)))))) / fma(3.0, Float64(1.0 + Float64(0.5 * fma(cos(x), t_2, t_0))), Float64(Float64(y * y) * fma(-0.75, t_0, Float64(0.0625 * Float64(Float64(y * y) * t_0))))));
	else
		tmp = Float64(Float64(fma(Float64(cos(x) - cos(y)), Float64(t_1 * sqrt(2.0)), 2.0) / 3.0) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.021], N[(N[(N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 0.021], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[(1.0 + N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(-0.75 * t$95$0 + N[(0.0625 * N[(N[(y * y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0210000000000000013

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 62.6

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

   6. Applied rewrites 62.6%

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

   if -0.0210000000000000013 < y < 0.0210000000000000013

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   10. Applied rewrites 49.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.75, 3 - \sqrt{5}, 0.0625 \cdot \left(\left(y \cdot y\right) \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   11. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 48.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.75, 3 - \sqrt{5}, 0.0625 \cdot \left(\left(y \cdot y\right) \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

   13. Applied rewrites 48.9%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\left(1 + -0.5 \cdot \left(y \cdot y\right)\right)}\right)}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.75, 3 - \sqrt{5}, 0.0625 \cdot \left(\left(y \cdot y\right) \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

   if 0.0210000000000000013 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sqrt.f64 62.6

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

   6. Applied rewrites 62.6%

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

Alternative 12: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.021:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(0.5 \cdot t\_0, \cos y, \mathsf{fma}\left(0.5 \cdot t\_1, \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 0.021:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot \left(y \cdot y\right)\right)\right)}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos x, t\_1, t\_0\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.75, t\_0, 0.0625 \cdot \left(\left(y \cdot y\right) \cdot t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (- (sqrt 5.0) 1.0)))
   (if (<= y -0.021)
     (/
      (/
       (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
       3.0)
      (fma (* 0.5 t_0) (cos y) (fma (* 0.5 t_1) (cos x) 1.0)))
     (if (<= y 0.021)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          (- (cos x) (+ 1.0 (* -0.5 (* y y))))))
        (fma
         3.0
         (+ 1.0 (* 0.5 (fma (cos x) t_1 t_0)))
         (* (* y y) (fma -0.75 t_0 (* 0.0625 (* (* y y) t_0))))))
       (/
        (+
         2.0
         (*
          (* (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y))))) (sqrt 2.0))
          (- (cos x) (cos y))))
        (*
         3.0
         (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.021) {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / fma((0.5 * t_0), cos(y), fma((0.5 * t_1), cos(x), 1.0));
	} else if (y <= 0.021) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * (cos(x) - (1.0 + (-0.5 * (y * y)))))) / fma(3.0, (1.0 + (0.5 * fma(cos(x), t_1, t_0))), ((y * y) * fma(-0.75, t_0, (0.0625 * ((y * y) * t_0)))));
	} else {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.021)
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / fma(Float64(0.5 * t_0), cos(y), fma(Float64(0.5 * t_1), cos(x), 1.0)));
	elseif (y <= 0.021)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * Float64(cos(x) - Float64(1.0 + Float64(-0.5 * Float64(y * y)))))) / fma(3.0, Float64(1.0 + Float64(0.5 * fma(cos(x), t_1, t_0))), Float64(Float64(y * y) * fma(-0.75, t_0, Float64(0.0625 * Float64(Float64(y * y) * t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.021], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.021], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[(1.0 + N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(-0.75 * t$95$0 + N[(0.0625 * N[(N[(y * y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0210000000000000013

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 62.6

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

   6. Applied rewrites 62.6%

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

   if -0.0210000000000000013 < y < 0.0210000000000000013

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

   10. Applied rewrites 49.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.75, 3 - \sqrt{5}, 0.0625 \cdot \left(\left(y \cdot y\right) \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   11. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 48.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(1 + -0.5 \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.75, 3 - \sqrt{5}, 0.0625 \cdot \left(\left(y \cdot y\right) \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

   13. Applied rewrites 48.9%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\left(1 + -0.5 \cdot \left(y \cdot y\right)\right)}\right)}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.75, 3 - \sqrt{5}, 0.0625 \cdot \left(\left(y \cdot y\right) \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

   if 0.0210000000000000013 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 62.6

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

   4. Applied rewrites 62.6%

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

Alternative 13: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 1 + \frac{t\_0}{2} \cdot \cos x\\ t_2 := 1 - 0.5 \cdot \left(y \cdot y\right)\\ t_3 := 3 - \sqrt{5}\\ t_4 := \frac{t\_3}{2}\\ \mathbf{if}\;y \leq -0.0058:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(0.5 \cdot t\_3, \cos y, \mathsf{fma}\left(0.5 \cdot t\_0, \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 0.0078:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - t\_2\right)}{3 \cdot \left(t\_1 + t\_4 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(t\_1 + t\_4 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (+ 1.0 (* (/ t_0 2.0) (cos x))))
        (t_2 (- 1.0 (* 0.5 (* y y))))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (/ t_3 2.0)))
   (if (<= y -0.0058)
     (/
      (/
       (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
       3.0)
      (fma (* 0.5 t_3) (cos y) (fma (* 0.5 t_0) (cos x) 1.0)))
     (if (<= y 0.0078)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          (- (cos x) t_2)))
        (* 3.0 (+ t_1 (* t_4 t_2))))
       (/
        (+
         2.0
         (*
          (* (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y))))) (sqrt 2.0))
          (- (cos x) (cos y))))
        (* 3.0 (+ t_1 (* t_4 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 1.0 + ((t_0 / 2.0) * cos(x));
	double t_2 = 1.0 - (0.5 * (y * y));
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 / 2.0;
	double tmp;
	if (y <= -0.0058) {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / fma((0.5 * t_3), cos(y), fma((0.5 * t_0), cos(x), 1.0));
	} else if (y <= 0.0078) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * (cos(x) - t_2))) / (3.0 * (t_1 + (t_4 * t_2)));
	} else {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * (t_1 + (t_4 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x)))
	t_2 = Float64(1.0 - Float64(0.5 * Float64(y * y)))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 / 2.0)
	tmp = 0.0
	if (y <= -0.0058)
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / fma(Float64(0.5 * t_3), cos(y), fma(Float64(0.5 * t_0), cos(x), 1.0)));
	elseif (y <= 0.0078)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * Float64(cos(x) - t_2))) / Float64(3.0 * Float64(t_1 + Float64(t_4 * t_2))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(t_1 + Float64(t_4 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / 2.0), $MachinePrecision]}, If[LessEqual[y, -0.0058], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[(0.5 * t$95$3), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0078], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0058

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 62.6

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

   6. Applied rewrites 62.6%

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

   if -0.0058 < y < 0.0077999999999999996

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 49.1

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

   10. Applied rewrites 49.1%

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

   11. Taylor expanded in y around 0

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

       1. fp-cancel-sign-sub-inv N/A

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

       2. metadata-eval N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 49.9

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

   13. Applied rewrites 49.9%

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

   if 0.0077999999999999996 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 62.6

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

   4. Applied rewrites 62.6%

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

Alternative 14: 79.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(0.5 \cdot t\_0, \cos y, \mathsf{fma}\left(0.5 \cdot t\_1, \cos x, 1\right)\right)}\\ t_3 := 1 - 0.5 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq -0.0058:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.0078:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - t\_3\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (/
           (fma
            (* -0.0625 (pow (sin y) 2.0))
            (* (- 1.0 (cos y)) (sqrt 2.0))
            2.0)
           3.0)
          (fma (* 0.5 t_0) (cos y) (fma (* 0.5 t_1) (cos x) 1.0))))
        (t_3 (- 1.0 (* 0.5 (* y y)))))
   (if (<= y -0.0058)
     t_2
     (if (<= y 0.0078)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          (- (cos x) t_3)))
        (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) t_3))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / fma((0.5 * t_0), cos(y), fma((0.5 * t_1), cos(x), 1.0));
	double t_3 = 1.0 - (0.5 * (y * y));
	double tmp;
	if (y <= -0.0058) {
		tmp = t_2;
	} else if (y <= 0.0078) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * (cos(x) - t_3))) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * t_3)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / fma(Float64(0.5 * t_0), cos(y), fma(Float64(0.5 * t_1), cos(x), 1.0)))
	t_3 = Float64(1.0 - Float64(0.5 * Float64(y * y)))
	tmp = 0.0
	if (y <= -0.0058)
		tmp = t_2;
	elseif (y <= 0.0078)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * Float64(cos(x) - t_3))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * t_3))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0058], t$95$2, If[LessEqual[y, 0.0078], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0058 or 0.0077999999999999996 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 62.6

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

   6. Applied rewrites 62.6%

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

   if -0.0058 < y < 0.0077999999999999996

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 49.1

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

   10. Applied rewrites 49.1%

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

   11. Taylor expanded in y around 0

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

       1. fp-cancel-sign-sub-inv N/A

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

       2. metadata-eval N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 49.9

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

   13. Applied rewrites 49.9%

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

Alternative 15: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(0.5 \cdot t\_0, \cos x, 1\right)\right)}\\ \mathbf{if}\;y \leq -30500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0014:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (/
           (fma
            (* -0.0625 (pow (sin y) 2.0))
            (* (- 1.0 (cos y)) (sqrt 2.0))
            2.0)
           3.0)
          (fma
           (* 0.5 (- 3.0 (sqrt 5.0)))
           (cos y)
           (fma (* 0.5 t_0) (cos x) 1.0)))))
   (if (<= y -30500.0)
     t_1
     (if (<= y 0.0014)
       (/
        (+
         2.0
         (*
          -0.0625
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (*
         3.0
         (+
          (+ 1.0 (* (/ t_0 2.0) (cos x)))
          (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / fma((0.5 * (3.0 - sqrt(5.0))), cos(y), fma((0.5 * t_0), cos(x), 1.0));
	double tmp;
	if (y <= -30500.0) {
		tmp = t_1;
	} else if (y <= 0.0014) {
		tmp = (2.0 + (-0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / fma(Float64(0.5 * Float64(3.0 - sqrt(5.0))), cos(y), fma(Float64(0.5 * t_0), cos(x), 1.0)))
	tmp = 0.0
	if (y <= -30500.0)
		tmp = t_1;
	elseif (y <= 0.0014)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * 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(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	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[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -30500.0], t$95$1, If[LessEqual[y, 0.0014], N[(N[(2.0 + N[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -30500 or 0.00139999999999999999 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 62.6

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

   6. Applied rewrites 62.6%

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

   if -30500 < y < 0.00139999999999999999

   1. Initial program 99.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.3

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 61.7%

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

Alternative 16: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - \cos y\right) \cdot \sqrt{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := 1 + \frac{t\_2}{2} \cdot \cos x\\ \mathbf{if}\;y \leq -30500:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), t\_0, 2\right)}{3 \cdot \left(t\_3 + \frac{t\_1}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.0014:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(t\_3 + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_0, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot t\_1, \cos y, \mathsf{fma}\left(0.5 \cdot t\_2, \cos x, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- 1.0 (cos y)) (sqrt 2.0)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (+ 1.0 (* (/ t_2 2.0) (cos x)))))
   (if (<= y -30500.0)
     (/
      (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y))))) t_0 2.0)
      (* 3.0 (+ t_3 (* (/ t_1 2.0) (cos y)))))
     (if (<= y 0.0014)
       (/
        (+
         2.0
         (*
          -0.0625
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (* 3.0 (+ t_3 (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))
       (/
        (* (fma (* -0.0625 (pow (sin y) 2.0)) t_0 2.0) 0.3333333333333333)
        (fma (* 0.5 t_1) (cos y) (fma (* 0.5 t_2) (cos x) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = (1.0 - cos(y)) * sqrt(2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 1.0 + ((t_2 / 2.0) * cos(x));
	double tmp;
	if (y <= -30500.0) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), t_0, 2.0) / (3.0 * (t_3 + ((t_1 / 2.0) * cos(y))));
	} else if (y <= 0.0014) {
		tmp = (2.0 + (-0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 * (t_3 + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	} else {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), t_0, 2.0) * 0.3333333333333333) / fma((0.5 * t_1), cos(y), fma((0.5 * t_2), cos(x), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 - cos(y)) * sqrt(2.0))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x)))
	tmp = 0.0
	if (y <= -30500.0)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), t_0, 2.0) / Float64(3.0 * Float64(t_3 + Float64(Float64(t_1 / 2.0) * cos(y)))));
	elseif (y <= 0.0014)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 * Float64(t_3 + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), t_0, 2.0) * 0.3333333333333333) / fma(Float64(0.5 * t_1), cos(y), fma(Float64(0.5 * t_2), cos(x), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -30500.0], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0014], N[(N[(2.0 + N[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -30500

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 62.6%

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

   if -30500 < y < 0.00139999999999999999

   1. Initial program 99.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.3

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 61.7%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

   if 0.00139999999999999999 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 62.5%

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

Alternative 17: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_0 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -30500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0014:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(t\_0 + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))))
        (t_1
         (/
          (fma
           (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (* 3.0 (+ t_0 (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))))
   (if (<= y -30500.0)
     t_1
     (if (<= y 0.0014)
       (/
        (+
         2.0
         (*
          -0.0625
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (* 3.0 (+ t_0 (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x));
	double t_1 = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (t_0 + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	double tmp;
	if (y <= -30500.0) {
		tmp = t_1;
	} else if (y <= 0.0014) {
		tmp = (2.0 + (-0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 * (t_0 + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)))
	t_1 = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(t_0 + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -30500.0)
		tmp = t_1;
	elseif (y <= 0.0014)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 * Float64(t_0 + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -30500.0], t$95$1, If[LessEqual[y, 0.0014], N[(N[(2.0 + N[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -30500 or 0.00139999999999999999 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 62.6%

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

   if -30500 < y < 0.00139999999999999999

   1. Initial program 99.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.3

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 61.7%

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

Alternative 18: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -30500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.0014:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\mathsf{fma}\left(0.5 \cdot t\_0, \cos y, \mathsf{fma}\left(t\_1 \cdot \cos x, 0.5, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (fma
           (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (*
           3.0
           (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))))
   (if (<= y -30500.0)
     t_2
     (if (<= y 0.0014)
       (/
        (*
         0.3333333333333333
         (-
          2.0
          (*
           0.0625
           (*
            (- 0.5 (* 0.5 (cos (* 2.0 x))))
            (* (sqrt 2.0) (- (cos x) 1.0))))))
        (fma (* 0.5 t_0) (cos y) (fma (* t_1 (cos x)) 0.5 1.0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	double tmp;
	if (y <= -30500.0) {
		tmp = t_2;
	} else if (y <= 0.0014) {
		tmp = (0.3333333333333333 * (2.0 - (0.0625 * ((0.5 - (0.5 * cos((2.0 * x)))) * (sqrt(2.0) * (cos(x) - 1.0)))))) / fma((0.5 * t_0), cos(y), fma((t_1 * cos(x)), 0.5, 1.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -30500.0)
		tmp = t_2;
	elseif (y <= 0.0014)
		tmp = Float64(Float64(0.3333333333333333 * Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))) / fma(Float64(0.5 * t_0), cos(y), fma(Float64(t_1 * cos(x)), 0.5, 1.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -30500.0], t$95$2, If[LessEqual[y, 0.0014], N[(N[(0.3333333333333333 * N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -30500 or 0.00139999999999999999 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 62.6%

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

   if -30500 < y < 0.00139999999999999999

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-cos.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 61.8%

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

Alternative 19: 79.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.00385:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_2, t\_1 \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{t\_3}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, \mathsf{fma}\left(t\_3, 0.5, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 - 0.0625 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot t\_1\right)\right)\right)}{\mathsf{fma}\left(0.5 \cdot t\_0, \cos y, \mathsf{fma}\left(t\_3 \cdot \cos x, 0.5, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) 1.0))
        (t_2 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.00385)
     (/
      (fma (* -0.0625 t_2) (* t_1 (sqrt 2.0)) 2.0)
      (* 3.0 (+ (+ 1.0 (* (/ t_3 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))
     (if (<= x 1.55e-6)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (* 1.5 (cos y)) t_0 (* (fma t_3 0.5 1.0) 3.0)))
       (/
        (* 0.3333333333333333 (- 2.0 (* 0.0625 (* t_2 (* (sqrt 2.0) t_1)))))
        (fma (* 0.5 t_0) (cos y) (fma (* t_3 (cos x)) 0.5 1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = 0.5 - (0.5 * cos((2.0 * x)));
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.00385) {
		tmp = fma((-0.0625 * t_2), (t_1 * sqrt(2.0)), 2.0) / (3.0 * ((1.0 + ((t_3 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	} else if (x <= 1.55e-6) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_0, (fma(t_3, 0.5, 1.0) * 3.0));
	} else {
		tmp = (0.3333333333333333 * (2.0 - (0.0625 * (t_2 * (sqrt(2.0) * t_1))))) / fma((0.5 * t_0), cos(y), fma((t_3 * cos(x)), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.00385)
		tmp = Float64(fma(Float64(-0.0625 * t_2), Float64(t_1 * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_3 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	elseif (x <= 1.55e-6)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(1.5 * cos(y)), t_0, Float64(fma(t_3, 0.5, 1.0) * 3.0)));
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(2.0 - Float64(0.0625 * Float64(t_2 * Float64(sqrt(2.0) * t_1))))) / fma(Float64(0.5 * t_0), cos(y), fma(Float64(t_3 * cos(x)), 0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.00385], N[(N[(N[(-0.0625 * t$95$2), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-6], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(t$95$3 * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(2.0 - N[(0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0038500000000000001

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 61.7%

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

   if -0.0038500000000000001 < x < 1.55e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.7%

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

   if 1.55e-6 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-cos.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 61.8%

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

Alternative 20: 79.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{0.3333333333333333 \cdot \left(2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\mathsf{fma}\left(0.5 \cdot t\_1, \cos y, \mathsf{fma}\left(t\_0 \cdot \cos x, 0.5, 1\right)\right)}\\ \mathbf{if}\;x \leq -0.00385:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_1, \mathsf{fma}\left(t\_0, 0.5, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (*
           0.3333333333333333
           (-
            2.0
            (*
             0.0625
             (*
              (- 0.5 (* 0.5 (cos (* 2.0 x))))
              (* (sqrt 2.0) (- (cos x) 1.0))))))
          (fma (* 0.5 t_1) (cos y) (fma (* t_0 (cos x)) 0.5 1.0)))))
   (if (<= x -0.00385)
     t_2
     (if (<= x 1.55e-6)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (* 1.5 (cos y)) t_1 (* (fma t_0 0.5 1.0) 3.0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (0.3333333333333333 * (2.0 - (0.0625 * ((0.5 - (0.5 * cos((2.0 * x)))) * (sqrt(2.0) * (cos(x) - 1.0)))))) / fma((0.5 * t_1), cos(y), fma((t_0 * cos(x)), 0.5, 1.0));
	double tmp;
	if (x <= -0.00385) {
		tmp = t_2;
	} else if (x <= 1.55e-6) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_1, (fma(t_0, 0.5, 1.0) * 3.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(0.3333333333333333 * Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))) / fma(Float64(0.5 * t_1), cos(y), fma(Float64(t_0 * cos(x)), 0.5, 1.0)))
	tmp = 0.0
	if (x <= -0.00385)
		tmp = t_2;
	elseif (x <= 1.55e-6)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(1.5 * cos(y)), t_1, Float64(fma(t_0, 0.5, 1.0) * 3.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.3333333333333333 * N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00385], t$95$2, If[LessEqual[x, 1.55e-6], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(t$95$0 * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0038500000000000001 or 1.55e-6 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-cos.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 61.8%

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

   if -0.0038500000000000001 < x < 1.55e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.7%

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

Alternative 21: 79.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(0.5 \cdot t\_2, \cos y, \mathsf{fma}\left(0.5 \cdot t\_0, \cos x, 1\right)\right)\\ \mathbf{if}\;x \leq -0.00385:\\ \;\;\;\;\frac{\frac{t\_1}{3}}{t\_3}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_2, \mathsf{fma}\left(t\_0, 0.5, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot 0.3333333333333333}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (fma
          (* (- (cos x) 1.0) (sqrt 2.0))
          (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
          2.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma (* 0.5 t_2) (cos y) (fma (* 0.5 t_0) (cos x) 1.0))))
   (if (<= x -0.00385)
     (/ (/ t_1 3.0) t_3)
     (if (<= x 1.55e-6)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (* 1.5 (cos y)) t_2 (* (fma t_0 0.5 1.0) 3.0)))
       (/ (* t_1 0.3333333333333333) t_3)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma((0.5 * t_2), cos(y), fma((0.5 * t_0), cos(x), 1.0));
	double tmp;
	if (x <= -0.00385) {
		tmp = (t_1 / 3.0) / t_3;
	} else if (x <= 1.55e-6) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_2, (fma(t_0, 0.5, 1.0) * 3.0));
	} else {
		tmp = (t_1 * 0.3333333333333333) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(Float64(0.5 * t_2), cos(y), fma(Float64(0.5 * t_0), cos(x), 1.0))
	tmp = 0.0
	if (x <= -0.00385)
		tmp = Float64(Float64(t_1 / 3.0) / t_3);
	elseif (x <= 1.55e-6)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(1.5 * cos(y)), t_2, Float64(fma(t_0, 0.5, 1.0) * 3.0)));
	else
		tmp = Float64(Float64(t_1 * 0.3333333333333333) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00385], N[(N[(t$95$1 / 3.0), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[x, 1.55e-6], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[(t$95$0 * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * 0.3333333333333333), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0038500000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 61.7%

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

   if -0.0038500000000000001 < x < 1.55e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.7%

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

   if 1.55e-6 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 61.7%

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

Alternative 22: 79.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot t\_1, \cos y, \mathsf{fma}\left(0.5 \cdot t\_0, \cos x, 1\right)\right)}\\ \mathbf{if}\;x \leq -0.00385:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_1, \mathsf{fma}\left(t\_0, 0.5, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (*
           (fma
            (* (- (cos x) 1.0) (sqrt 2.0))
            (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
            2.0)
           0.3333333333333333)
          (fma (* 0.5 t_1) (cos y) (fma (* 0.5 t_0) (cos x) 1.0)))))
   (if (<= x -0.00385)
     t_2
     (if (<= x 1.55e-6)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (* 1.5 (cos y)) t_1 (* (fma t_0 0.5 1.0) 3.0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0) * 0.3333333333333333) / fma((0.5 * t_1), cos(y), fma((0.5 * t_0), cos(x), 1.0));
	double tmp;
	if (x <= -0.00385) {
		tmp = t_2;
	} else if (x <= 1.55e-6) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_1, (fma(t_0, 0.5, 1.0) * 3.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0) * 0.3333333333333333) / fma(Float64(0.5 * t_1), cos(y), fma(Float64(0.5 * t_0), cos(x), 1.0)))
	tmp = 0.0
	if (x <= -0.00385)
		tmp = t_2;
	elseif (x <= 1.55e-6)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(1.5 * cos(y)), t_1, Float64(fma(t_0, 0.5, 1.0) * 3.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00385], t$95$2, If[LessEqual[x, 1.55e-6], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(t$95$0 * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0038500000000000001 or 1.55e-6 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 61.7%

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

   if -0.0038500000000000001 < x < 1.55e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.7%

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

Alternative 23: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00385:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(1.5, t\_2, \mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_2, \mathsf{fma}\left(t\_0, 0.5, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_2\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (fma
          (* (- (cos x) 1.0) (sqrt 2.0))
          (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
          2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.00385)
     (/ t_1 (fma 1.5 t_2 (* (fma (* 0.5 (cos x)) t_0 1.0) 3.0)))
     (if (<= x 2.9e-6)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (* 1.5 (cos y)) t_2 (* (fma t_0 0.5 1.0) 3.0)))
       (* (/ t_1 (fma (fma t_0 (cos x) t_2) 0.5 1.0)) 0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.00385) {
		tmp = t_1 / fma(1.5, t_2, (fma((0.5 * cos(x)), t_0, 1.0) * 3.0));
	} else if (x <= 2.9e-6) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_2, (fma(t_0, 0.5, 1.0) * 3.0));
	} else {
		tmp = (t_1 / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.00385)
		tmp = Float64(t_1 / fma(1.5, t_2, Float64(fma(Float64(0.5 * cos(x)), t_0, 1.0) * 3.0)));
	elseif (x <= 2.9e-6)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(1.5 * cos(y)), t_2, Float64(fma(t_0, 0.5, 1.0) * 3.0)));
	else
		tmp = Float64(Float64(t_1 / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00385], N[(t$95$1 / N[(1.5 * t$95$2 + N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-6], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[(t$95$0 * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0038500000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 59.4%

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

   if -0.0038500000000000001 < x < 2.9000000000000002e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 59.7%

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

   if 2.9000000000000002e-6 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.4%

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

   6. Applied rewrites 59.4%

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

Alternative 24: 78.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00385:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(1.5, t\_2, \mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_2\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (fma
          (* (- (cos x) 1.0) (sqrt 2.0))
          (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
          2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.00385)
     (/ t_1 (fma 1.5 t_2 (* (fma (* 0.5 (cos x)) t_0 1.0) 3.0)))
     (if (<= x 2.9e-6)
       (*
        (/
         (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
         (fma 0.5 (fma t_2 (cos y) t_0) 1.0))
        0.3333333333333333)
       (* (/ t_1 (fma (fma t_0 (cos x) t_2) 0.5 1.0)) 0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.00385) {
		tmp = t_1 / fma(1.5, t_2, (fma((0.5 * cos(x)), t_0, 1.0) * 3.0));
	} else if (x <= 2.9e-6) {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (t_1 / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.00385)
		tmp = Float64(t_1 / fma(1.5, t_2, Float64(fma(Float64(0.5 * cos(x)), t_0, 1.0) * 3.0)));
	elseif (x <= 2.9e-6)
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(t_1 / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00385], N[(t$95$1 / N[(1.5 * t$95$2 + N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-6], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(t$95$1 / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0038500000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 59.4%

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

   if -0.0038500000000000001 < x < 2.9000000000000002e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 59.6%

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

   if 2.9000000000000002e-6 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.4%

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

   6. Applied rewrites 59.4%

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

Alternative 25: 78.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00385:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(1.5, t\_2, \mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_2\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (fma
          (* (- (cos x) 1.0) (sqrt 2.0))
          (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
          2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.00385)
     (/ t_1 (fma 1.5 t_2 (* (fma (* 0.5 (cos x)) t_0 1.0) 3.0)))
     (if (<= x 2.9e-6)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_2 (cos y) t_0) 1.0))
        0.3333333333333333)
       (* (/ t_1 (fma (fma t_0 (cos x) t_2) 0.5 1.0)) 0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.00385) {
		tmp = t_1 / fma(1.5, t_2, (fma((0.5 * cos(x)), t_0, 1.0) * 3.0));
	} else if (x <= 2.9e-6) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (t_1 / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.00385)
		tmp = Float64(t_1 / fma(1.5, t_2, Float64(fma(Float64(0.5 * cos(x)), t_0, 1.0) * 3.0)));
	elseif (x <= 2.9e-6)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(t_1 / fma(fma(t_0, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00385], N[(t$95$1 / N[(1.5 * t$95$2 + N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-6], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(t$95$1 / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0038500000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 59.4%

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

   if -0.0038500000000000001 < x < 2.9000000000000002e-6

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.6%

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

   if 2.9000000000000002e-6 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.4%

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

   6. Applied rewrites 59.4%

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

Alternative 26: 78.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00385:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_0, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_2\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.00385)
     (*
      (/
       (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_0 2.0)
       (fma 0.5 (- (fma t_1 (cos x) 3.0) (sqrt 5.0)) 1.0))
      0.3333333333333333)
     (if (<= x 2.9e-6)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_2 (cos y) t_1) 1.0))
        0.3333333333333333)
       (*
        (/
         (fma t_0 (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625) 2.0)
         (fma (fma t_1 (cos x) t_2) 0.5 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = (cos(x) - 1.0) * sqrt(2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.00385) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), t_0, 2.0) / fma(0.5, (fma(t_1, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	} else if (x <= 2.9e-6) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_1), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (fma(t_0, ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0) / fma(fma(t_1, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.00385)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), t_0, 2.0) / fma(0.5, Float64(fma(t_1, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333);
	elseif (x <= 2.9e-6)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_1), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(t_0, Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0) / fma(fma(t_1, cos(x), t_2), 0.5, 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00385], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 2.9e-6], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(t$95$0 * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0038500000000000001

   1. Initial program 99.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.4%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift-cos.f64 59.4

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

   6. Applied rewrites 59.4%

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

   if -0.0038500000000000001 < x < 2.9000000000000002e-6

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.6%

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

   if 2.9000000000000002e-6 < x

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.4%

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

   6. Applied rewrites 59.4%

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

Alternative 27: 59.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 59.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. associate-+r- N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lift-cos.f64 59.4

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

6. Applied rewrites 59.4%

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

Alternative 28: 59.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 59.4%

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

6. Applied rewrites 59.4%

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

Alternative 29: 59.4% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (*
   (fma
    (* (- (cos x) 1.0) (sqrt 2.0))
    (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
    2.0)
   0.3333333333333333)
  (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) {
	return (fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 0.5, 1.0);
}

Julia

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

Wolfram

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 59.4%

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

6. Applied rewrites 59.4%

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

Alternative 30: 42.5% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 59.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 42.5%

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

Alternative 31: 40.0% accurate, 316.7× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 59.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 40.0%

   \[\leadsto 0.3333333333333333 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025136 
(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))))))