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

Percentage Accurate: 99.3% → 99.3%

Time: 31.0s

Alternatives: 40

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 99.3

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

9. Applied rewrites 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

7. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (/
   (fma
    (*
     (* (- (sin x) (* 0.0625 (sin y))) (sqrt 2.0))
     (- (sin y) (* 0.0625 (sin x))))
    (- (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) - (0.0625 * sin(y))) * sqrt(2.0)) * (sin(y) - (0.0625 * sin(x)))), (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(Float64(fma(Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sqrt(2.0)) * Float64(sin(y) - Float64(0.0625 * sin(x)))), Float64(cos(x) - cos(y)), 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)) / 3.0)
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 99.3

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

9. Applied rewrites 99.3%

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

10. Taylor expanded in x around inf

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

12. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 99.3%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (* (sqrt 2.0) (- (cos x) (cos y)))
    (* (- (sin y) (* 0.0625 (sin x))) (- (sin x) (* 0.0625 (sin y))))
    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((sqrt(2.0) * (cos(x) - cos(y))), ((sin(y) - (0.0625 * sin(x))) * (sin(x) - (0.0625 * sin(y)))), 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(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 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[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 99.2%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 99.2%

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

Alternative 6: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sin x - \frac{\sin y}{16}\\ t_2 := \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)\\ \mathbf{if}\;y \leq -0.75:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot t\_1\right) \cdot \sin y\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{5} \cdot \sqrt{5} - 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.95:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), t\_0, 2\right)}{t\_2}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y \cdot \left(t\_1 \cdot \sqrt{2}\right), t\_0, 2\right)}{t\_2 \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sin x) (/ (sin y) 16.0)))
        (t_2
         (fma
          (cos y)
          (/ (- 3.0 (sqrt 5.0)) 2.0)
          (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))))
   (if (<= y -0.75)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) t_1) (sin y)) t_0))
      (*
       3.0
       (+
        (+
         1.0
         (*
          (/ (/ (- (* (sqrt 5.0) (sqrt 5.0)) 1.0) (+ (sqrt 5.0) 1.0)) 2.0)
          (cos x)))
        (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))
     (if (<= y 0.95)
       (/
        (/
         (fma
          (*
           (*
            (-
             (sin x)
             (*
              (fma
               (-
                (*
                 (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
                 (* y y))
                0.010416666666666666)
               (* y y)
               0.0625)
              y))
            (sqrt 2.0))
           (- (sin y) (* 0.0625 (sin x))))
          t_0
          2.0)
         t_2)
        3.0)
       (/ (fma (* (sin y) (* t_1 (sqrt 2.0))) t_0 2.0) (* t_2 3.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sin(x) - (sin(y) / 16.0);
	double t_2 = fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0));
	double tmp;
	if (y <= -0.75) {
		tmp = (2.0 + (((sqrt(2.0) * t_1) * sin(y)) * t_0)) / (3.0 * ((1.0 + (((((sqrt(5.0) * sqrt(5.0)) - 1.0) / (sqrt(5.0) + 1.0)) / 2.0) * cos(x))) + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	} else if (y <= 0.95) {
		tmp = (fma((((sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y)) * sqrt(2.0)) * (sin(y) - (0.0625 * sin(x)))), t_0, 2.0) / t_2) / 3.0;
	} else {
		tmp = fma((sin(y) * (t_1 * sqrt(2.0))), t_0, 2.0) / (t_2 * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sin(x) - Float64(sin(y) / 16.0))
	t_2 = fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0))
	tmp = 0.0
	if (y <= -0.75)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * t_1) * sin(y)) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(Float64(Float64(sqrt(5.0) * sqrt(5.0)) - 1.0) / Float64(sqrt(5.0) + 1.0)) / 2.0) * cos(x))) + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	elseif (y <= 0.95)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y)) * sqrt(2.0)) * Float64(sin(y) - Float64(0.0625 * sin(x)))), t_0, 2.0) / t_2) / 3.0);
	else
		tmp = Float64(fma(Float64(sin(y) * Float64(t_1 * sqrt(2.0))), t_0, 2.0) / Float64(t_2 * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.75], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 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], If[LessEqual[y, 0.95], N[(N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[Sin[y], $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(t$95$2 * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.75

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 60.8

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

   5. Applied rewrites 60.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 60.8

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

   7. Applied rewrites 60.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 60.8

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

   9. Applied rewrites 60.8%

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

   if -0.75 < y < 0.94999999999999996

   1. Initial program 99.5%

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 99.6

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\sin x - y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\sin x - \left(\frac{1}{16} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) - \frac{1}{96}\right)\right) \cdot y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\sin x - \left(\frac{1}{16} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) - \frac{1}{96}\right)\right) \cdot y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3} \]

   if 0.94999999999999996 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 51.8

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

   5. Applied rewrites 51.8%

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

   7. Applied rewrites 51.9%

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

Alternative 7: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)\\ \mathbf{if}\;y \leq -0.75 \lor \neg \left(y \leq 0.95\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), t\_0, 2\right)}{t\_1 \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), t\_0, 2\right)}{t\_1}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (fma
          (cos y)
          (/ (- 3.0 (sqrt 5.0)) 2.0)
          (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))))
   (if (or (<= y -0.75) (not (<= y 0.95)))
     (/
      (fma (* (sin y) (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0))) t_0 2.0)
      (* t_1 3.0))
     (/
      (/
       (fma
        (*
         (*
          (-
           (sin x)
           (*
            (fma
             (-
              (*
               (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
               (* y y))
              0.010416666666666666)
             (* y y)
             0.0625)
            y))
          (sqrt 2.0))
         (- (sin y) (* 0.0625 (sin x))))
        t_0
        2.0)
       t_1)
      3.0))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0));
	double tmp;
	if ((y <= -0.75) || !(y <= 0.95)) {
		tmp = fma((sin(y) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), t_0, 2.0) / (t_1 * 3.0);
	} else {
		tmp = (fma((((sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y)) * sqrt(2.0)) * (sin(y) - (0.0625 * sin(x)))), t_0, 2.0) / t_1) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0))
	tmp = 0.0
	if ((y <= -0.75) || !(y <= 0.95))
		tmp = Float64(fma(Float64(sin(y) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), t_0, 2.0) / Float64(t_1 * 3.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y)) * sqrt(2.0)) * Float64(sin(y) - Float64(0.0625 * sin(x)))), t_0, 2.0) / t_1) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.75], N[Not[LessEqual[y, 0.95]], $MachinePrecision]], N[(N[(N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(t$95$1 * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] / 3.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.75 or 0.94999999999999996 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 56.5

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

   5. Applied rewrites 56.5%

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

   7. Applied rewrites 56.5%

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

   if -0.75 < y < 0.94999999999999996

   1. Initial program 99.5%

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 99.6

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\sin x - y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\sin x - \left(\frac{1}{16} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) - \frac{1}{96}\right)\right) \cdot y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\sin x - \left(\frac{1}{16} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) - \frac{1}{96}\right)\right) \cdot y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.75 \lor \neg \left(y \leq 0.95\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \sin x \cdot \sqrt{2}\\ t_3 := \mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)\\ \mathbf{if}\;x \leq -0.32:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\sin y - \frac{\sin x}{16}\right) \cdot t\_2, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, \frac{4}{\left(1 + \sqrt{5}\right) \cdot 2}, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.205:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\sin y - \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right) \cdot x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{t\_3 \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2 \cdot \left(\sin y - 0.0625 \cdot \sin x\right), t\_1, 2\right)}{t\_3}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (* (sin x) (sqrt 2.0)))
        (t_3 (fma (cos y) t_0 (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))))
   (if (<= x -0.32)
     (/
      (fma t_1 (* (- (sin y) (/ (sin x) 16.0)) t_2) 2.0)
      (*
       (fma (cos y) t_0 (fma (cos x) (/ 4.0 (* (+ 1.0 (sqrt 5.0)) 2.0)) 1.0))
       3.0))
     (if (<= x 0.205)
       (/
        (fma
         t_1
         (*
          (- (sin y) (* (fma -0.010416666666666666 (* x x) 0.0625) x))
          (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
         2.0)
        (* t_3 3.0))
       (/ (/ (fma (* t_2 (- (sin y) (* 0.0625 (sin x)))) t_1 2.0) t_3) 3.0)))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = sin(x) * sqrt(2.0);
	double t_3 = fma(cos(y), t_0, fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0));
	double tmp;
	if (x <= -0.32) {
		tmp = fma(t_1, ((sin(y) - (sin(x) / 16.0)) * t_2), 2.0) / (fma(cos(y), t_0, fma(cos(x), (4.0 / ((1.0 + sqrt(5.0)) * 2.0)), 1.0)) * 3.0);
	} else if (x <= 0.205) {
		tmp = fma(t_1, ((sin(y) - (fma(-0.010416666666666666, (x * x), 0.0625) * x)) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 2.0) / (t_3 * 3.0);
	} else {
		tmp = (fma((t_2 * (sin(y) - (0.0625 * sin(x)))), t_1, 2.0) / t_3) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sin(x) * sqrt(2.0))
	t_3 = fma(cos(y), t_0, fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0))
	tmp = 0.0
	if (x <= -0.32)
		tmp = Float64(fma(t_1, Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * t_2), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), Float64(4.0 / Float64(Float64(1.0 + sqrt(5.0)) * 2.0)), 1.0)) * 3.0));
	elseif (x <= 0.205)
		tmp = Float64(fma(t_1, Float64(Float64(sin(y) - Float64(fma(-0.010416666666666666, Float64(x * x), 0.0625) * x)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), 2.0) / Float64(t_3 * 3.0));
	else
		tmp = Float64(Float64(fma(Float64(t_2 * Float64(sin(y) - Float64(0.0625 * sin(x)))), t_1, 2.0) / t_3) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.32], N[(N[(t$95$1 * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(4.0 / N[(N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.205], N[(N[(t$95$1 * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + 0.0625), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$3 * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.320000000000000007

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. lift-sin.f64 52.3

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

   7. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

   9. Applied rewrites 52.4%

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

   if -0.320000000000000007 < x < 0.204999999999999988

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right) \cdot x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. lower-*.f64 99.8

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right) \cdot x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   7. Applied rewrites 99.8%

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

   if 0.204999999999999988 < x

   1. Initial program 98.7%

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

   4. Applied rewrites 98.9%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 99.1

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in y around 0

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

       1. lift-sin.f64 62.7

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

   12. Applied rewrites 62.7%

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

Alternative 9: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5} - 1}{2}\\ t_1 := \frac{3 - \sqrt{5}}{2}\\ \mathbf{if}\;y \leq -0.38 \lor \neg \left(y \leq 0.66\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, t\_1, \mathsf{fma}\left(\cos x, t\_0, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)\right)}{3 \cdot \left(\left(1 + t\_0 \cdot \cos x\right) + t\_1 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt 5.0) 1.0) 2.0)) (t_1 (/ (- 3.0 (sqrt 5.0)) 2.0)))
   (if (or (<= y -0.38) (not (<= y 0.66)))
     (/
      (fma
       (* (sin y) (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
       (- (cos x) (cos y))
       2.0)
      (* (fma (cos y) t_1 (fma (cos x) t_0 1.0)) 3.0))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (fma
           (-
            (*
             (fma (* y y) -0.0005208333333333333 0.010416666666666666)
             (* y y))
            0.0625)
           y
           (sin x)))
         (- (sin y) (/ (sin x) 16.0)))
        (-
         (cos x)
         (fma
          (-
           (* (fma (* y y) -0.001388888888888889 0.041666666666666664) (* y y))
           0.5)
          (* y y)
          1.0))))
      (* 3.0 (+ (+ 1.0 (* t_0 (cos x))) (* t_1 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = (sqrt(5.0) - 1.0) / 2.0;
	double t_1 = (3.0 - sqrt(5.0)) / 2.0;
	double tmp;
	if ((y <= -0.38) || !(y <= 0.66)) {
		tmp = fma((sin(y) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), (cos(x) - cos(y)), 2.0) / (fma(cos(y), t_1, fma(cos(x), t_0, 1.0)) * 3.0);
	} else {
		tmp = (2.0 + (((sqrt(2.0) * fma(((fma((y * y), -0.0005208333333333333, 0.010416666666666666) * (y * y)) - 0.0625), y, sin(x))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - fma(((fma((y * y), -0.001388888888888889, 0.041666666666666664) * (y * y)) - 0.5), (y * y), 1.0)))) / (3.0 * ((1.0 + (t_0 * cos(x))) + (t_1 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	tmp = 0.0
	if ((y <= -0.38) || !(y <= 0.66))
		tmp = Float64(fma(Float64(sin(y) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), Float64(cos(x) - cos(y)), 2.0) / Float64(fma(cos(y), t_1, fma(cos(x), t_0, 1.0)) * 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(Float64(Float64(fma(Float64(y * y), -0.0005208333333333333, 0.010416666666666666) * Float64(y * y)) - 0.0625), y, sin(x))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - fma(Float64(Float64(fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664) * Float64(y * y)) - 0.5), Float64(y * y), 1.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_0 * cos(x))) + Float64(t_1 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.38], N[Not[LessEqual[y, 0.66]], $MachinePrecision]], N[(N[(N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.0005208333333333333 + 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.38 or 0.660000000000000031 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 56.5

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

   5. Applied rewrites 56.5%

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

   7. Applied rewrites 56.5%

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

   if -0.38 < y < 0.660000000000000031

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(\sin x + y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right)\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) + \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) \cdot y + \sin \color{blue}{x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, \color{blue}{y}, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {y}^{2} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left({y}^{2} \cdot \frac{-1}{1920} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-sin.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) \cdot {y}^{2} + 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, \color{blue}{{y}^{2}}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, {\color{blue}{y}}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left({y}^{2} \cdot \frac{-1}{720} + \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{720}, \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(y \cdot y\right) - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(y \cdot y\right) - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot \color{blue}{y}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lift-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot \color{blue}{y}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.38 \lor \neg \left(y \leq 0.66\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), \cos x - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\\ \mathbf{if}\;y \leq -0.38 \lor \neg \left(y \leq 0.66\right):\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(t\_1 + \left(0.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)\right)}{3 \cdot \left(t\_1 + \frac{t\_0}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))))
   (if (or (<= y -0.38) (not (<= y 0.66)))
     (/
      (+
       2.0
       (*
        (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y))
        (- (cos x) (cos y))))
      (* 3.0 (+ t_1 (* (* 0.5 (cos y)) t_0))))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (fma
           (-
            (*
             (fma (* y y) -0.0005208333333333333 0.010416666666666666)
             (* y y))
            0.0625)
           y
           (sin x)))
         (- (sin y) (/ (sin x) 16.0)))
        (-
         (cos x)
         (fma
          (-
           (* (fma (* y y) -0.001388888888888889 0.041666666666666664) (* y y))
           0.5)
          (* y y)
          1.0))))
      (* 3.0 (+ t_1 (* (/ t_0 2.0) (cos y))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x));
	double tmp;
	if ((y <= -0.38) || !(y <= 0.66)) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * (cos(x) - cos(y)))) / (3.0 * (t_1 + ((0.5 * cos(y)) * t_0)));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * fma(((fma((y * y), -0.0005208333333333333, 0.010416666666666666) * (y * y)) - 0.0625), y, sin(x))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - fma(((fma((y * y), -0.001388888888888889, 0.041666666666666664) * (y * y)) - 0.5), (y * y), 1.0)))) / (3.0 * (t_1 + ((t_0 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)))
	tmp = 0.0
	if ((y <= -0.38) || !(y <= 0.66))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(t_1 + Float64(Float64(0.5 * cos(y)) * t_0))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(Float64(Float64(fma(Float64(y * y), -0.0005208333333333333, 0.010416666666666666) * Float64(y * y)) - 0.0625), y, sin(x))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - fma(Float64(Float64(fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664) * Float64(y * y)) - 0.5), Float64(y * y), 1.0)))) / Float64(3.0 * Float64(t_1 + 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[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.38], N[Not[LessEqual[y, 0.66]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.0005208333333333333 + 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.38 or 0.660000000000000031 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. lift-sin.f64 56.5

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 56.5

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

   8. Applied rewrites 56.5%

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

   if -0.38 < y < 0.660000000000000031

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(\sin x + y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right)\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) + \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) \cdot y + \sin \color{blue}{x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, \color{blue}{y}, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {y}^{2} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left({y}^{2} \cdot \frac{-1}{1920} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-sin.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) \cdot {y}^{2} + 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, \color{blue}{{y}^{2}}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, {\color{blue}{y}}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left({y}^{2} \cdot \frac{-1}{720} + \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{720}, \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(y \cdot y\right) - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(y \cdot y\right) - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot \color{blue}{y}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lift-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot \color{blue}{y}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.38 \lor \neg \left(y \leq 0.66\right):\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := \mathsf{fma}\left(t\_0, t\_2 \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)\\ t_4 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.014:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_4 \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.031:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_2 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_4\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(0.5 \cdot \cos x, t\_4, 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sin y) (/ (sin x) 16.0)))
        (t_3 (fma t_0 (* t_2 (* (sin x) (sqrt 2.0))) 2.0))
        (t_4 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.014)
     (/ t_3 (* (fma 0.5 (fma t_1 (cos y) (* t_4 (cos x))) 1.0) 3.0))
     (if (<= x 0.031)
       (/
        (fma
         t_0
         (*
          t_2
          (*
           (fma (fma (* x x) -0.16666666666666666 1.0) x (* -0.0625 (sin y)))
           (sqrt 2.0)))
         2.0)
        (fma
         (fma 0.5 (fma t_1 (cos y) t_4) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_4)))
       (/
        t_3
        (* (fma (cos y) (/ t_1 2.0) (fma (* 0.5 (cos x)) t_4 1.0)) 3.0))))))

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 = sin(y) - (sin(x) / 16.0);
	double t_3 = fma(t_0, (t_2 * (sin(x) * sqrt(2.0))), 2.0);
	double t_4 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.014) {
		tmp = t_3 / (fma(0.5, fma(t_1, cos(y), (t_4 * cos(x))), 1.0) * 3.0);
	} else if (x <= 0.031) {
		tmp = fma(t_0, (t_2 * (fma(fma((x * x), -0.16666666666666666, 1.0), x, (-0.0625 * sin(y))) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(t_1, cos(y), t_4), 1.0), 3.0, ((-0.75 * (x * x)) * t_4));
	} else {
		tmp = t_3 / (fma(cos(y), (t_1 / 2.0), fma((0.5 * cos(x)), t_4, 1.0)) * 3.0);
	}
	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(sin(y) - Float64(sin(x) / 16.0))
	t_3 = fma(t_0, Float64(t_2 * Float64(sin(x) * sqrt(2.0))), 2.0)
	t_4 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.014)
		tmp = Float64(t_3 / Float64(fma(0.5, fma(t_1, cos(y), Float64(t_4 * cos(x))), 1.0) * 3.0));
	elseif (x <= 0.031)
		tmp = Float64(fma(t_0, Float64(t_2 * Float64(fma(fma(Float64(x * x), -0.16666666666666666, 1.0), x, Float64(-0.0625 * sin(y))) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(t_1, cos(y), t_4), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_4)));
	else
		tmp = Float64(t_3 / Float64(fma(cos(y), Float64(t_1 / 2.0), fma(Float64(0.5 * cos(x)), t_4, 1.0)) * 3.0));
	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[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.014], N[(t$95$3 / N[(N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(t$95$4 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.031], N[(N[(t$95$0 * N[(t$95$2 * N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$4), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0140000000000000003

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. lift-sin.f64 52.3

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

   7. Applied rewrites 52.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 52.3%

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

   if -0.0140000000000000003 < x < 0.031

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{1}{16} \cdot \sin y\right)} \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin y}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \frac{-1}{16} \cdot \sin \color{blue}{y}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(1 + \frac{-1}{6} \cdot {x}^{2}, \color{blue}{x}, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{6} \cdot {x}^{2} + 1, x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \frac{-1}{6} + 1, x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sin.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)} \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 0.031 < x

   1. Initial program 98.7%

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.5

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

   7. Applied rewrites 62.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 62.5%

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

Alternative 12: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.014 \lor \neg \left(x \leq 0.031\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_0 \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_1 \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_0 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.014) (not (<= x 0.031)))
     (/
      (fma t_2 (* t_0 (* (sin x) (sqrt 2.0))) 2.0)
      (* (fma 0.5 (fma t_3 (cos y) (* t_1 (cos x))) 1.0) 3.0))
     (/
      (fma
       t_2
       (*
        t_0
        (*
         (fma (fma (* x x) -0.16666666666666666 1.0) x (* -0.0625 (sin y)))
         (sqrt 2.0)))
       2.0)
      (fma
       (fma 0.5 (fma t_3 (cos y) t_1) 1.0)
       3.0
       (* (* -0.75 (* x x)) t_1))))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.014) || !(x <= 0.031)) {
		tmp = fma(t_2, (t_0 * (sin(x) * sqrt(2.0))), 2.0) / (fma(0.5, fma(t_3, cos(y), (t_1 * cos(x))), 1.0) * 3.0);
	} else {
		tmp = fma(t_2, (t_0 * (fma(fma((x * x), -0.16666666666666666, 1.0), x, (-0.0625 * sin(y))) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(t_3, cos(y), t_1), 1.0), 3.0, ((-0.75 * (x * x)) * t_1));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.014) || !(x <= 0.031))
		tmp = Float64(fma(t_2, Float64(t_0 * Float64(sin(x) * sqrt(2.0))), 2.0) / Float64(fma(0.5, fma(t_3, cos(y), Float64(t_1 * cos(x))), 1.0) * 3.0));
	else
		tmp = Float64(fma(t_2, Float64(t_0 * Float64(fma(fma(Float64(x * x), -0.16666666666666666, 1.0), x, Float64(-0.0625 * sin(y))) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(t_3, cos(y), t_1), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.014], N[Not[LessEqual[x, 0.031]], $MachinePrecision]], N[(N[(t$95$2 * N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(t$95$0 * N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0140000000000000003 or 0.031 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in y around 0

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

      1. lift-sin.f64 57.5

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

   7. Applied rewrites 57.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 57.4%

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

   if -0.0140000000000000003 < x < 0.031

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{1}{16} \cdot \sin y\right)} \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin y}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \frac{-1}{16} \cdot \sin \color{blue}{y}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(1 + \frac{-1}{6} \cdot {x}^{2}, \color{blue}{x}, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{6} \cdot {x}^{2} + 1, x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \frac{-1}{6} + 1, x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sin.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)} \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.014 \lor \neg \left(x \leq 0.031\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{\sqrt{5} - 1}{2}\\ \mathbf{if}\;y \leq -0.38 \lor \neg \left(y \leq 0.85\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_1, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)\right)}{3 \cdot \left(\left(1 + t\_1 \cdot \cos x\right) + t\_0 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0)) (t_1 (/ (- (sqrt 5.0) 1.0) 2.0)))
   (if (or (<= y -0.38) (not (<= y 0.85)))
     (/
      (fma
       (- (cos x) (cos y))
       (* (* -0.0625 (pow (sin y) 2.0)) (sqrt 2.0))
       2.0)
      (* (fma (cos y) t_0 (fma (cos x) t_1 1.0)) 3.0))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (fma
           (-
            (*
             (fma (* y y) -0.0005208333333333333 0.010416666666666666)
             (* y y))
            0.0625)
           y
           (sin x)))
         (- (sin y) (/ (sin x) 16.0)))
        (-
         (cos x)
         (fma
          (-
           (* (fma (* y y) -0.001388888888888889 0.041666666666666664) (* y y))
           0.5)
          (* y y)
          1.0))))
      (* 3.0 (+ (+ 1.0 (* t_1 (cos x))) (* t_0 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = (sqrt(5.0) - 1.0) / 2.0;
	double tmp;
	if ((y <= -0.38) || !(y <= 0.85)) {
		tmp = fma((cos(x) - cos(y)), ((-0.0625 * pow(sin(y), 2.0)) * sqrt(2.0)), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_1, 1.0)) * 3.0);
	} else {
		tmp = (2.0 + (((sqrt(2.0) * fma(((fma((y * y), -0.0005208333333333333, 0.010416666666666666) * (y * y)) - 0.0625), y, sin(x))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - fma(((fma((y * y), -0.001388888888888889, 0.041666666666666664) * (y * y)) - 0.5), (y * y), 1.0)))) / (3.0 * ((1.0 + (t_1 * cos(x))) + (t_0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	tmp = 0.0
	if ((y <= -0.38) || !(y <= 0.85))
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), t_1, 1.0)) * 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(Float64(Float64(fma(Float64(y * y), -0.0005208333333333333, 0.010416666666666666) * Float64(y * y)) - 0.0625), y, sin(x))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - fma(Float64(Float64(fma(Float64(y * y), -0.001388888888888889, 0.041666666666666664) * Float64(y * y)) - 0.5), Float64(y * y), 1.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_1 * cos(x))) + Float64(t_0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.38], N[Not[LessEqual[y, 0.85]], $MachinePrecision]], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.0005208333333333333 + 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.38 or 0.849999999999999978 < y

   1. Initial program 98.9%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sqrt.f64 52.9

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

   7. Applied rewrites 52.9%

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

   if -0.38 < y < 0.849999999999999978

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(\sin x + y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right)\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) + \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) \cdot y + \sin \color{blue}{x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, \color{blue}{y}, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {y}^{2} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left({y}^{2} \cdot \frac{-1}{1920} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-sin.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \left(\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right) \cdot {y}^{2} + 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, \color{blue}{{y}^{2}}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}, {\color{blue}{y}}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {y}^{2} + \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\left({y}^{2} \cdot \frac{-1}{720} + \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{720}, \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot {y}^{2} - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(y \cdot y\right) - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(y \cdot y\right) - \frac{1}{2}, {y}^{2}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(y \cdot y\right) - \frac{1}{2}, y \cdot \color{blue}{y}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lift-*.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot \color{blue}{y}, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.38 \lor \neg \left(y \leq 0.85\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(y \cdot y\right) - 0.5, y \cdot y, 1\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ \mathbf{if}\;y \leq -0.38 \lor \neg \left(y \leq 0.66\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_2, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + t\_2 \cdot \cos x\right) + t\_0 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0)))
   (if (or (<= y -0.38) (not (<= y 0.66)))
     (/
      (fma t_1 (* (* -0.0625 (pow (sin y) 2.0)) (sqrt 2.0)) 2.0)
      (* (fma (cos y) t_0 (fma (cos x) t_2 1.0)) 3.0))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (fma
           (-
            (*
             (fma (* y y) -0.0005208333333333333 0.010416666666666666)
             (* y y))
            0.0625)
           y
           (sin x)))
         (fma
          (fma
           (- (* (* y y) 0.008333333333333333) 0.16666666666666666)
           (* y y)
           1.0)
          y
          (* -0.0625 (sin x))))
        t_1))
      (* 3.0 (+ (+ 1.0 (* t_2 (cos x))) (* t_0 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double tmp;
	if ((y <= -0.38) || !(y <= 0.66)) {
		tmp = fma(t_1, ((-0.0625 * pow(sin(y), 2.0)) * sqrt(2.0)), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0);
	} else {
		tmp = (2.0 + (((sqrt(2.0) * fma(((fma((y * y), -0.0005208333333333333, 0.010416666666666666) * (y * y)) - 0.0625), y, sin(x))) * fma(fma((((y * y) * 0.008333333333333333) - 0.16666666666666666), (y * y), 1.0), y, (-0.0625 * sin(x)))) * t_1)) / (3.0 * ((1.0 + (t_2 * cos(x))) + (t_0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	tmp = 0.0
	if ((y <= -0.38) || !(y <= 0.66))
		tmp = Float64(fma(t_1, Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(Float64(Float64(fma(Float64(y * y), -0.0005208333333333333, 0.010416666666666666) * Float64(y * y)) - 0.0625), y, sin(x))) * fma(fma(Float64(Float64(Float64(y * y) * 0.008333333333333333) - 0.16666666666666666), Float64(y * y), 1.0), y, Float64(-0.0625 * sin(x)))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * cos(x))) + Float64(t_0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.38], N[Not[LessEqual[y, 0.66]], $MachinePrecision]], N[(N[(t$95$1 * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.0005208333333333333 + 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.38 or 0.660000000000000031 < y

   1. Initial program 98.9%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sqrt.f64 52.9

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

   7. Applied rewrites 52.9%

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

   if -0.38 < y < 0.660000000000000031

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(\sin x + y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right)\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) + \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) \cdot y + \sin \color{blue}{x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, \color{blue}{y}, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {y}^{2} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left({y}^{2} \cdot \frac{-1}{1920} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-sin.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right), \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lift-sin.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.38 \lor \neg \left(y \leq 0.66\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ \mathbf{if}\;y \leq -0.2 \lor \neg \left(y \leq 0.17\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_2, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + t\_2 \cdot \cos x\right) + t\_0 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0)))
   (if (or (<= y -0.2) (not (<= y 0.17)))
     (/
      (fma t_1 (* (* -0.0625 (pow (sin y) 2.0)) (sqrt 2.0)) 2.0)
      (* (fma (cos y) t_0 (fma (cos x) t_2 1.0)) 3.0))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (fma
           (-
            (*
             (fma (* y y) -0.0005208333333333333 0.010416666666666666)
             (* y y))
            0.0625)
           y
           (sin x)))
         (fma (fma (* y y) -0.16666666666666666 1.0) y (* -0.0625 (sin x))))
        t_1))
      (* 3.0 (+ (+ 1.0 (* t_2 (cos x))) (* t_0 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double tmp;
	if ((y <= -0.2) || !(y <= 0.17)) {
		tmp = fma(t_1, ((-0.0625 * pow(sin(y), 2.0)) * sqrt(2.0)), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0);
	} else {
		tmp = (2.0 + (((sqrt(2.0) * fma(((fma((y * y), -0.0005208333333333333, 0.010416666666666666) * (y * y)) - 0.0625), y, sin(x))) * fma(fma((y * y), -0.16666666666666666, 1.0), y, (-0.0625 * sin(x)))) * t_1)) / (3.0 * ((1.0 + (t_2 * cos(x))) + (t_0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	tmp = 0.0
	if ((y <= -0.2) || !(y <= 0.17))
		tmp = Float64(fma(t_1, Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(Float64(Float64(fma(Float64(y * y), -0.0005208333333333333, 0.010416666666666666) * Float64(y * y)) - 0.0625), y, sin(x))) * fma(fma(Float64(y * y), -0.16666666666666666, 1.0), y, Float64(-0.0625 * sin(x)))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * cos(x))) + Float64(t_0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.2], N[Not[LessEqual[y, 0.17]], $MachinePrecision]], N[(N[(t$95$1 * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.0005208333333333333 + 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.20000000000000001 or 0.170000000000000012 < y

   1. Initial program 98.9%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sqrt.f64 52.9

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

   7. Applied rewrites 52.9%

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

   if -0.20000000000000001 < y < 0.170000000000000012

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(\sin x + y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right)\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) + \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) \cdot y + \sin \color{blue}{x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, \color{blue}{y}, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {y}^{2} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left({y}^{2} \cdot \frac{-1}{1920} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-sin.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {y}^{2}, \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \frac{-1}{6} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-sin.f64 99.2

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.2 \lor \neg \left(y \leq 0.17\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 0.0285\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_2, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(y - 0.0625 \cdot \sin x\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + t\_2 \cdot \cos x\right) + t\_0 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0)))
   (if (or (<= y -0.14) (not (<= y 0.0285)))
     (/
      (fma t_1 (* (* -0.0625 (pow (sin y) 2.0)) (sqrt 2.0)) 2.0)
      (* (fma (cos y) t_0 (fma (cos x) t_2 1.0)) 3.0))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (fma
           (-
            (*
             (fma (* y y) -0.0005208333333333333 0.010416666666666666)
             (* y y))
            0.0625)
           y
           (sin x)))
         (- y (* 0.0625 (sin x))))
        t_1))
      (* 3.0 (+ (+ 1.0 (* t_2 (cos x))) (* t_0 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double tmp;
	if ((y <= -0.14) || !(y <= 0.0285)) {
		tmp = fma(t_1, ((-0.0625 * pow(sin(y), 2.0)) * sqrt(2.0)), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0);
	} else {
		tmp = (2.0 + (((sqrt(2.0) * fma(((fma((y * y), -0.0005208333333333333, 0.010416666666666666) * (y * y)) - 0.0625), y, sin(x))) * (y - (0.0625 * sin(x)))) * t_1)) / (3.0 * ((1.0 + (t_2 * cos(x))) + (t_0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	tmp = 0.0
	if ((y <= -0.14) || !(y <= 0.0285))
		tmp = Float64(fma(t_1, Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), t_2, 1.0)) * 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(Float64(Float64(fma(Float64(y * y), -0.0005208333333333333, 0.010416666666666666) * Float64(y * y)) - 0.0625), y, sin(x))) * Float64(y - Float64(0.0625 * sin(x)))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * cos(x))) + Float64(t_0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.14], N[Not[LessEqual[y, 0.0285]], $MachinePrecision]], N[(N[(t$95$1 * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.0005208333333333333 + 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.14000000000000001 or 0.028500000000000001 < y

   1. Initial program 98.9%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sqrt.f64 52.9

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

   7. Applied rewrites 52.9%

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

   if -0.14000000000000001 < y < 0.028500000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(\sin x + y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right)\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) + \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) \cdot y + \sin \color{blue}{x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, \color{blue}{y}, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{-1}{1920} \cdot {y}^{2} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left({y}^{2} \cdot \frac{-1}{1920} + \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-sin.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \color{blue}{\left(y - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(y - \color{blue}{\frac{1}{16} \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{1920}, \frac{1}{96}\right) \cdot \left(y \cdot y\right) - \frac{1}{16}, y, \sin x\right)\right) \cdot \left(y - \frac{1}{16} \cdot \color{blue}{\sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-sin.f64 98.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(y - 0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 98.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \color{blue}{\left(y - 0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 0.0285\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right) \cdot \left(y \cdot y\right) - 0.0625, y, \sin x\right)\right) \cdot \left(y - 0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{t\_0}{2}\\ t_2 := \cos x - \cos y\\ t_3 := 3 - \sqrt{5}\\ t_4 := \frac{t\_3}{2}\\ \mathbf{if}\;x \leq -0.014:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right) \cdot t\_2}{3 \cdot \left(\left(1 + t\_1 \cdot \cos x\right) + t\_4 \cdot \cos y\right)}\\ \mathbf{elif}\;x \leq 0.031:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_0\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_4, \mathsf{fma}\left(\cos x, t\_1, 1\right)\right)}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (/ t_0 2.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (/ t_3 2.0)))
   (if (<= x -0.014)
     (/
      (+ 2.0 (* (* (* -0.0625 (pow (sin x) 2.0)) (sqrt 2.0)) t_2))
      (* 3.0 (+ (+ 1.0 (* t_1 (cos x))) (* t_4 (cos y)))))
     (if (<= x 0.031)
       (/
        (fma
         t_2
         (*
          (- (sin y) (/ (sin x) 16.0))
          (*
           (fma (fma (* x x) -0.16666666666666666 1.0) x (* -0.0625 (sin y)))
           (sqrt 2.0)))
         2.0)
        (fma
         (fma 0.5 (fma t_3 (cos y) t_0) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_0)))
       (/
        (/
         (fma
          (* (- (cos x) 1.0) (sqrt 2.0))
          (* (- 0.5 (* 0.5 (cos (* 2.0 x)))) -0.0625)
          2.0)
         (fma (cos y) t_4 (fma (cos x) t_1 1.0)))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = t_0 / 2.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 / 2.0;
	double tmp;
	if (x <= -0.014) {
		tmp = (2.0 + (((-0.0625 * pow(sin(x), 2.0)) * sqrt(2.0)) * t_2)) / (3.0 * ((1.0 + (t_1 * cos(x))) + (t_4 * cos(y))));
	} else if (x <= 0.031) {
		tmp = fma(t_2, ((sin(y) - (sin(x) / 16.0)) * (fma(fma((x * x), -0.16666666666666666, 1.0), x, (-0.0625 * sin(y))) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(t_3, cos(y), t_0), 1.0), 3.0, ((-0.75 * (x * x)) * t_0));
	} else {
		tmp = (fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (0.5 * cos((2.0 * x)))) * -0.0625), 2.0) / fma(cos(y), t_4, fma(cos(x), t_1, 1.0))) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(t_0 / 2.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 / 2.0)
	tmp = 0.0
	if (x <= -0.014)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * sqrt(2.0)) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_1 * cos(x))) + Float64(t_4 * cos(y)))));
	elseif (x <= 0.031)
		tmp = Float64(fma(t_2, Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(fma(fma(Float64(x * x), -0.16666666666666666, 1.0), x, Float64(-0.0625 * sin(y))) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(t_3, cos(y), t_0), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_0)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * -0.0625), 2.0) / fma(cos(y), t_4, fma(cos(x), t_1, 1.0))) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $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[x, -0.014], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.031], N[(N[(t$95$2 * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * t$95$4 + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0140000000000000003

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sqrt.f64 48.1

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

   5. Applied rewrites 48.1%

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

   if -0.0140000000000000003 < x < 0.031

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{1}{16} \cdot \sin y\right)} \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin y}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \frac{-1}{16} \cdot \sin \color{blue}{y}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(1 + \frac{-1}{6} \cdot {x}^{2}, \color{blue}{x}, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{6} \cdot {x}^{2} + 1, x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left({x}^{2} \cdot \frac{-1}{6} + 1, x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right), x, \frac{-1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sin.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right)} \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 0.031 < x

   1. Initial program 98.7%

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

   4. Applied rewrites 98.9%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

   9. Applied rewrites 58.4%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 58.4

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

   11. Applied rewrites 58.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.014:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;x \leq 0.031:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right), x, -0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{t\_0}{2}\\ t_2 := \cos x - \cos y\\ t_3 := 3 - \sqrt{5}\\ t_4 := \frac{t\_3}{2}\\ \mathbf{if}\;x \leq -0.014:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right) \cdot t\_2}{3 \cdot \left(\left(1 + t\_1 \cdot \cos x\right) + t\_4 \cdot \cos y\right)}\\ \mathbf{elif}\;x \leq 0.031:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \left(\sin y - \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right) \cdot x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_0\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_4, \mathsf{fma}\left(\cos x, t\_1, 1\right)\right)}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (/ t_0 2.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (/ t_3 2.0)))
   (if (<= x -0.014)
     (/
      (+ 2.0 (* (* (* -0.0625 (pow (sin x) 2.0)) (sqrt 2.0)) t_2))
      (* 3.0 (+ (+ 1.0 (* t_1 (cos x))) (* t_4 (cos y)))))
     (if (<= x 0.031)
       (/
        (fma
         t_2
         (*
          (- (sin y) (* (fma -0.010416666666666666 (* x x) 0.0625) x))
          (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
         2.0)
        (fma
         (fma 0.5 (fma t_3 (cos y) t_0) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_0)))
       (/
        (/
         (fma
          (* (- (cos x) 1.0) (sqrt 2.0))
          (* (- 0.5 (* 0.5 (cos (* 2.0 x)))) -0.0625)
          2.0)
         (fma (cos y) t_4 (fma (cos x) t_1 1.0)))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = t_0 / 2.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 / 2.0;
	double tmp;
	if (x <= -0.014) {
		tmp = (2.0 + (((-0.0625 * pow(sin(x), 2.0)) * sqrt(2.0)) * t_2)) / (3.0 * ((1.0 + (t_1 * cos(x))) + (t_4 * cos(y))));
	} else if (x <= 0.031) {
		tmp = fma(t_2, ((sin(y) - (fma(-0.010416666666666666, (x * x), 0.0625) * x)) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(t_3, cos(y), t_0), 1.0), 3.0, ((-0.75 * (x * x)) * t_0));
	} else {
		tmp = (fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (0.5 * cos((2.0 * x)))) * -0.0625), 2.0) / fma(cos(y), t_4, fma(cos(x), t_1, 1.0))) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(t_0 / 2.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 / 2.0)
	tmp = 0.0
	if (x <= -0.014)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * sqrt(2.0)) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_1 * cos(x))) + Float64(t_4 * cos(y)))));
	elseif (x <= 0.031)
		tmp = Float64(fma(t_2, Float64(Float64(sin(y) - Float64(fma(-0.010416666666666666, Float64(x * x), 0.0625) * x)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(t_3, cos(y), t_0), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_0)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * -0.0625), 2.0) / fma(cos(y), t_4, fma(cos(x), t_1, 1.0))) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $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[x, -0.014], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.031], N[(N[(t$95$2 * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + 0.0625), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * t$95$4 + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0140000000000000003

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sqrt.f64 48.1

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

   5. Applied rewrites 48.1%

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

   if -0.0140000000000000003 < x < 0.031

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \left(\frac{-1}{96} \cdot {x}^{2} + \frac{1}{16}\right) \cdot x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \mathsf{fma}\left(\frac{-1}{96}, {x}^{2}, \frac{1}{16}\right) \cdot x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right) \cdot x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-*.f64 99.5

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right) \cdot x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \color{blue}{\mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right) \cdot x}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 0.031 < x

   1. Initial program 98.7%

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

   4. Applied rewrites 98.9%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

   9. Applied rewrites 58.4%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 58.4

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

   11. Applied rewrites 58.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.014:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;x \leq 0.031:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right) \cdot x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{t\_0}{2}\\ t_2 := \cos x - \cos y\\ t_3 := 3 - \sqrt{5}\\ t_4 := \frac{t\_3}{2}\\ \mathbf{if}\;x \leq -0.014:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right) \cdot t\_2}{3 \cdot \left(\left(1 + t\_1 \cdot \cos x\right) + t\_4 \cdot \cos y\right)}\\ \mathbf{elif}\;x \leq 0.031:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_0\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_4, \mathsf{fma}\left(\cos x, t\_1, 1\right)\right)}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (/ t_0 2.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (/ t_3 2.0)))
   (if (<= x -0.014)
     (/
      (+ 2.0 (* (* (* -0.0625 (pow (sin x) 2.0)) (sqrt 2.0)) t_2))
      (* 3.0 (+ (+ 1.0 (* t_1 (cos x))) (* t_4 (cos y)))))
     (if (<= x 0.031)
       (/
        (fma
         t_2
         (*
          (fma -0.0625 x (sin y))
          (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
         2.0)
        (fma
         (fma 0.5 (fma t_3 (cos y) t_0) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_0)))
       (/
        (/
         (fma
          (* (- (cos x) 1.0) (sqrt 2.0))
          (* (- 0.5 (* 0.5 (cos (* 2.0 x)))) -0.0625)
          2.0)
         (fma (cos y) t_4 (fma (cos x) t_1 1.0)))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = t_0 / 2.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 / 2.0;
	double tmp;
	if (x <= -0.014) {
		tmp = (2.0 + (((-0.0625 * pow(sin(x), 2.0)) * sqrt(2.0)) * t_2)) / (3.0 * ((1.0 + (t_1 * cos(x))) + (t_4 * cos(y))));
	} else if (x <= 0.031) {
		tmp = fma(t_2, (fma(-0.0625, x, sin(y)) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(t_3, cos(y), t_0), 1.0), 3.0, ((-0.75 * (x * x)) * t_0));
	} else {
		tmp = (fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (0.5 * cos((2.0 * x)))) * -0.0625), 2.0) / fma(cos(y), t_4, fma(cos(x), t_1, 1.0))) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(t_0 / 2.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 / 2.0)
	tmp = 0.0
	if (x <= -0.014)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * sqrt(2.0)) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_1 * cos(x))) + Float64(t_4 * cos(y)))));
	elseif (x <= 0.031)
		tmp = Float64(fma(t_2, Float64(fma(-0.0625, x, sin(y)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(t_3, cos(y), t_0), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_0)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * -0.0625), 2.0) / fma(cos(y), t_4, fma(cos(x), t_1, 1.0))) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $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[x, -0.014], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.031], N[(N[(t$95$2 * N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * t$95$4 + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0140000000000000003

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sqrt.f64 48.1

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

   5. Applied rewrites 48.1%

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

   if -0.0140000000000000003 < x < 0.031

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 99.4

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

   10. Applied rewrites 99.4%

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

   if 0.031 < x

   1. Initial program 98.7%

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

   4. Applied rewrites 98.9%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

   9. Applied rewrites 58.4%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 58.4

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

   11. Applied rewrites 58.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.014:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;x \leq 0.031:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 15500\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - 1, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= y -0.14) (not (<= y 15500.0)))
     (/
      (fma
       (- (cos x) (cos y))
       (* (* -0.0625 (pow (sin y) 2.0)) (sqrt 2.0))
       2.0)
      (* (fma (cos y) (/ t_1 2.0) (fma (cos x) (/ t_0 2.0) 1.0)) 3.0))
     (/
      (fma
       (- (cos x) 1.0)
       (*
        (- (sin y) (/ (sin x) 16.0))
        (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
       2.0)
      (* (fma (fma t_0 (cos x) t_1) 0.5 1.0) 3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((y <= -0.14) || !(y <= 15500.0)) {
		tmp = fma((cos(x) - cos(y)), ((-0.0625 * pow(sin(y), 2.0)) * sqrt(2.0)), 2.0) / (fma(cos(y), (t_1 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0)) * 3.0);
	} else {
		tmp = fma((cos(x) - 1.0), ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 2.0) / (fma(fma(t_0, cos(x), t_1), 0.5, 1.0) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((y <= -0.14) || !(y <= 15500.0))
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)) * 3.0));
	else
		tmp = Float64(fma(Float64(cos(x) - 1.0), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), 2.0) / Float64(fma(fma(t_0, cos(x), t_1), 0.5, 1.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.14], N[Not[LessEqual[y, 15500.0]], $MachinePrecision]], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.14000000000000001 or 15500 < y

   1. Initial program 98.9%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sqrt.f64 53.1

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

   7. Applied rewrites 53.1%

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

   if -0.14000000000000001 < y < 15500

   1. Initial program 99.5%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 97.7%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 15500\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - 1, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 0.0007\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \left(y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (cos x) (cos y))))
   (if (or (<= y -0.14) (not (<= y 0.0007)))
     (/
      (fma t_2 (* (* -0.0625 (pow (sin y) 2.0)) (sqrt 2.0)) 2.0)
      (* (fma (cos y) (/ t_1 2.0) (fma (cos x) (/ t_0 2.0) 1.0)) 3.0))
     (/
      (fma
       t_2
       (* (- y (* 0.0625 (sin x))) (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
       2.0)
      (* (fma (fma t_0 (cos x) t_1) 0.5 1.0) 3.0)))))

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 = cos(x) - cos(y);
	double tmp;
	if ((y <= -0.14) || !(y <= 0.0007)) {
		tmp = fma(t_2, ((-0.0625 * pow(sin(y), 2.0)) * sqrt(2.0)), 2.0) / (fma(cos(y), (t_1 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0)) * 3.0);
	} else {
		tmp = fma(t_2, ((y - (0.0625 * sin(x))) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 2.0) / (fma(fma(t_0, cos(x), t_1), 0.5, 1.0) * 3.0);
	}
	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(cos(x) - cos(y))
	tmp = 0.0
	if ((y <= -0.14) || !(y <= 0.0007))
		tmp = Float64(fma(t_2, Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)) * 3.0));
	else
		tmp = Float64(fma(t_2, Float64(Float64(y - Float64(0.0625 * sin(x))) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), 2.0) / Float64(fma(fma(t_0, cos(x), t_1), 0.5, 1.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.14], N[Not[LessEqual[y, 0.0007]], $MachinePrecision]], N[(N[(t$95$2 * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[(y - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.14000000000000001 or 6.99999999999999993e-4 < y

   1. Initial program 98.9%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sqrt.f64 52.9

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

   7. Applied rewrites 52.9%

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

   if -0.14000000000000001 < y < 6.99999999999999993e-4

   1. Initial program 99.5%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 98.3

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

   10. Applied rewrites 98.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 0.0007\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 0.0007\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= y -0.14) (not (<= y 0.0007)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (* (fma (cos y) (/ t_1 2.0) (fma (cos x) (/ t_0 2.0) 1.0)) 3.0))
     (/
      (fma
       (- (cos x) (cos y))
       (* (- y (* 0.0625 (sin x))) (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
       2.0)
      (* (fma (fma t_0 (cos x) t_1) 0.5 1.0) 3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((y <= -0.14) || !(y <= 0.0007)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(cos(y), (t_1 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0)) * 3.0);
	} else {
		tmp = fma((cos(x) - cos(y)), ((y - (0.0625 * sin(x))) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 2.0) / (fma(fma(t_0, cos(x), t_1), 0.5, 1.0) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((y <= -0.14) || !(y <= 0.0007))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)) * 3.0));
	else
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(y - Float64(0.0625 * sin(x))) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), 2.0) / Float64(fma(fma(t_0, cos(x), t_1), 0.5, 1.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.14], N[Not[LessEqual[y, 0.0007]], $MachinePrecision]], 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[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(y - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.14000000000000001 or 6.99999999999999993e-4 < y

   1. Initial program 98.9%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

   7. Applied rewrites 52.8%

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

   if -0.14000000000000001 < y < 6.99999999999999993e-4

   1. Initial program 99.5%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 98.3

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

   10. Applied rewrites 98.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 0.0007\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 0.0007\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= y -0.14) (not (<= y 0.0007)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (* (fma (cos y) (/ t_1 2.0) (fma (cos x) (/ t_0 2.0) 1.0)) 3.0))
     (/
      (fma
       (- (cos x) (cos y))
       (* (- (sin y) (/ (sin x) 16.0)) (* (fma -0.0625 y (sin x)) (sqrt 2.0)))
       2.0)
      (* (fma (fma t_0 (cos x) t_1) 0.5 1.0) 3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((y <= -0.14) || !(y <= 0.0007)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(cos(y), (t_1 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0)) * 3.0);
	} else {
		tmp = fma((cos(x) - cos(y)), ((sin(y) - (sin(x) / 16.0)) * (fma(-0.0625, y, sin(x)) * sqrt(2.0))), 2.0) / (fma(fma(t_0, cos(x), t_1), 0.5, 1.0) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((y <= -0.14) || !(y <= 0.0007))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)) * 3.0));
	else
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0))), 2.0) / Float64(fma(fma(t_0, cos(x), t_1), 0.5, 1.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.14], N[Not[LessEqual[y, 0.0007]], $MachinePrecision]], 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[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.14000000000000001 or 6.99999999999999993e-4 < y

   1. Initial program 98.9%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

   7. Applied rewrites 52.8%

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

   if -0.14000000000000001 < y < 6.99999999999999993e-4

   1. Initial program 99.5%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 98.3

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

   10. Applied rewrites 98.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 0.0007\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 15500\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= y -0.14) (not (<= y 15500.0)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (* (fma (cos y) (/ t_1 2.0) (fma (cos x) (/ t_0 2.0) 1.0)) 3.0))
     (/
      (fma
       (- (cos x) (cos y))
       (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
       2.0)
      (* (fma (fma t_0 (cos x) t_1) 0.5 1.0) 3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((y <= -0.14) || !(y <= 15500.0)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(cos(y), (t_1 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0)) * 3.0);
	} else {
		tmp = fma((cos(x) - cos(y)), ((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))), 2.0) / (fma(fma(t_0, cos(x), t_1), 0.5, 1.0) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((y <= -0.14) || !(y <= 15500.0))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)) * 3.0));
	else
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))), 2.0) / Float64(fma(fma(t_0, cos(x), t_1), 0.5, 1.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.14], N[Not[LessEqual[y, 15500.0]], $MachinePrecision]], 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[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.14000000000000001 or 15500 < y

   1. Initial program 98.9%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

   7. Applied rewrites 53.1%

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

   if -0.14000000000000001 < y < 15500

   1. Initial program 99.5%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. lift-sin.f64 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 97.5%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 15500\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_3 := \mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right)\\ \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2, {\sin x}^{2} \cdot -0.0625, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right), 1\right)}}{3}\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{t\_3 \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot -0.0625, 2\right)}{t\_3}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_3 (fma (cos y) (/ t_1 2.0) (fma (cos x) (/ t_0 2.0) 1.0))))
   (if (<= x -0.00065)
     (/
      (/
       (fma t_2 (* (pow (sin x) 2.0) -0.0625) 2.0)
       (fma 0.5 (fma t_0 (cos x) (* t_1 (cos y))) 1.0))
      3.0)
     (if (<= x 1.46e-28)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (* t_3 3.0))
       (/
        (/ (fma t_2 (* (- 0.5 (* 0.5 (cos (* 2.0 x)))) -0.0625) 2.0) t_3)
        3.0)))))

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 = (cos(x) - 1.0) * sqrt(2.0);
	double t_3 = fma(cos(y), (t_1 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0));
	double tmp;
	if (x <= -0.00065) {
		tmp = (fma(t_2, (pow(sin(x), 2.0) * -0.0625), 2.0) / fma(0.5, fma(t_0, cos(x), (t_1 * cos(y))), 1.0)) / 3.0;
	} else if (x <= 1.46e-28) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (t_3 * 3.0);
	} else {
		tmp = (fma(t_2, ((0.5 - (0.5 * cos((2.0 * x)))) * -0.0625), 2.0) / t_3) / 3.0;
	}
	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(cos(x) - 1.0) * sqrt(2.0))
	t_3 = fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0))
	tmp = 0.0
	if (x <= -0.00065)
		tmp = Float64(Float64(fma(t_2, Float64((sin(x) ^ 2.0) * -0.0625), 2.0) / fma(0.5, fma(t_0, cos(x), Float64(t_1 * cos(y))), 1.0)) / 3.0);
	elseif (x <= 1.46e-28)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(t_3 * 3.0));
	else
		tmp = Float64(Float64(fma(t_2, Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * -0.0625), 2.0) / t_3) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00065], N[(N[(N[(t$95$2 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[x, 1.46e-28], 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[(t$95$3 * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.4999999999999997e-4

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

   9. Applied rewrites 47.8%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 47.9%

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

   if -6.4999999999999997e-4 < x < 1.46e-28

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

   7. Applied rewrites 99.7%

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

   if 1.46e-28 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 99.0%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

   9. Applied rewrites 60.8%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 60.8

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

   11. Applied rewrites 60.8%

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

Alternative 26: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 0.0012\right):\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_1 \cdot \cos x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right)}{\mathsf{fma}\left(-0.25 \cdot \left(y \cdot y\right), t\_0, \mathsf{fma}\left(t\_1, \cos x, t\_0\right) \cdot 0.5\right) + 1}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (- (sqrt 5.0) 1.0)))
   (if (or (<= y -0.14) (not (<= y 0.0012)))
     (/
      (/
       (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
       3.0)
      (fma 0.5 (fma t_0 (cos y) (* t_1 (cos x))) 1.0))
     (/
      (/
       (fma (* (- (cos x) 1.0) (sqrt 2.0)) (* (pow (sin x) 2.0) -0.0625) 2.0)
       (+ (fma (* -0.25 (* y y)) t_0 (* (fma t_1 (cos x) t_0) 0.5)) 1.0))
      3.0))))

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 <= -0.14) || !(y <= 0.0012)) {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / 3.0) / fma(0.5, fma(t_0, cos(y), (t_1 * cos(x))), 1.0);
	} else {
		tmp = (fma(((cos(x) - 1.0) * sqrt(2.0)), (pow(sin(x), 2.0) * -0.0625), 2.0) / (fma((-0.25 * (y * y)), t_0, (fma(t_1, cos(x), t_0) * 0.5)) + 1.0)) / 3.0;
	}
	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 <= -0.14) || !(y <= 0.0012))
		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(0.5, fma(t_0, cos(y), Float64(t_1 * cos(x))), 1.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64((sin(x) ^ 2.0) * -0.0625), 2.0) / Float64(fma(Float64(-0.25 * Float64(y * y)), t_0, Float64(fma(t_1, cos(x), t_0) * 0.5)) + 1.0)) / 3.0);
	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[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.14], N[Not[LessEqual[y, 0.0012]], $MachinePrecision]], 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[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(-0.25 * N[(y * y), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.14000000000000001 or 0.00119999999999999989 < y

   1. Initial program 98.9%

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

   7. Applied rewrites 52.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 52.6%

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

   if -0.14000000000000001 < y < 0.00119999999999999989

   1. Initial program 99.5%

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

   9. Applied rewrites 98.0%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right)}{\color{blue}{\mathsf{fma}\left(-0.25 \cdot \left(y \cdot y\right), 3 - \sqrt{5}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) \cdot 0.5\right) + 1}}}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.14 \lor \neg \left(y \leq 0.0012\right):\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3}}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right)}{\mathsf{fma}\left(-0.25 \cdot \left(y \cdot y\right), 3 - \sqrt{5}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) \cdot 0.5\right) + 1}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, {\sin x}^{2} \cdot -0.0625, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2 \cdot \cos y\right), 1\right)}}{3}\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_2}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right)}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -9.8e-8)
     (/
      (/
       (fma t_1 (* (pow (sin x) 2.0) -0.0625) 2.0)
       (fma 0.5 (fma t_0 (cos x) (* t_2 (cos y))) 1.0))
      3.0)
     (if (<= x 1.46e-28)
       (/
        (/
         (fma (* (* (pow (sin y) 2.0) -0.0625) (- 1.0 (cos y))) (sqrt 2.0) 2.0)
         (fma 0.5 (fma t_2 (cos y) t_0) 1.0))
        3.0)
       (/
        (/
         (fma t_1 (* (- 0.5 (* 0.5 (cos (* 2.0 x)))) -0.0625) 2.0)
         (fma (cos y) (/ t_2 2.0) (fma (cos x) (/ t_0 2.0) 1.0)))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -9.8e-8) {
		tmp = (fma(t_1, (pow(sin(x), 2.0) * -0.0625), 2.0) / fma(0.5, fma(t_0, cos(x), (t_2 * cos(y))), 1.0)) / 3.0;
	} else if (x <= 1.46e-28) {
		tmp = (fma(((pow(sin(y), 2.0) * -0.0625) * (1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) / 3.0;
	} else {
		tmp = (fma(t_1, ((0.5 - (0.5 * cos((2.0 * x)))) * -0.0625), 2.0) / fma(cos(y), (t_2 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0))) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -9.8e-8)
		tmp = Float64(Float64(fma(t_1, Float64((sin(x) ^ 2.0) * -0.0625), 2.0) / fma(0.5, fma(t_0, cos(x), Float64(t_2 * cos(y))), 1.0)) / 3.0);
	elseif (x <= 1.46e-28)
		tmp = Float64(Float64(fma(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) / 3.0);
	else
		tmp = Float64(Float64(fma(t_1, Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * -0.0625), 2.0) / fma(cos(y), Float64(t_2 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0))) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.8e-8], N[(N[(N[(t$95$1 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[x, 1.46e-28], N[(N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(t$95$1 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8000000000000004e-8

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

   9. Applied rewrites 48.5%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 48.6%

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

   if -9.8000000000000004e-8 < x < 1.46e-28

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.6%

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

   if 1.46e-28 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 99.0%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

   9. Applied rewrites 60.8%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 60.8

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

   11. Applied rewrites 60.8%

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

Alternative 28: 78.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-8} \lor \neg \left(x \leq 1.46 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right)}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -9.8e-8) (not (<= x 1.46e-28)))
     (/
      (/
       (fma
        (* (- (cos x) 1.0) (sqrt 2.0))
        (* (- 0.5 (* 0.5 (cos (* 2.0 x)))) -0.0625)
        2.0)
       (fma (cos y) (/ t_1 2.0) (fma (cos x) (/ t_0 2.0) 1.0)))
      3.0)
     (/
      (/
       (fma (* (* (pow (sin y) 2.0) -0.0625) (- 1.0 (cos y))) (sqrt 2.0) 2.0)
       (fma 0.5 (fma t_1 (cos y) t_0) 1.0))
      3.0))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -9.8e-8) || !(x <= 1.46e-28)) {
		tmp = (fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (0.5 * cos((2.0 * x)))) * -0.0625), 2.0) / fma(cos(y), (t_1 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0))) / 3.0;
	} else {
		tmp = (fma(((pow(sin(y), 2.0) * -0.0625) * (1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -9.8e-8) || !(x <= 1.46e-28))
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * -0.0625), 2.0) / fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0))) / 3.0);
	else
		tmp = Float64(Float64(fma(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -9.8e-8], N[Not[LessEqual[x, 1.46e-28]], $MachinePrecision]], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.8000000000000004e-8 or 1.46e-28 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

   9. Applied rewrites 54.9%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 54.2

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

   11. Applied rewrites 54.2%

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

   if -9.8000000000000004e-8 < x < 1.46e-28

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.6%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-8} \lor \neg \left(x \leq 1.46 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 78.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ t_3 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_2, t\_3, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_3, t\_2 \cdot -0.0625, 2\right)}{\mathsf{fma}\left(1, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right)}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (pow (sin x) 2.0))
        (t_3 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -9.8e-8)
     (*
      (/ (fma (* -0.0625 t_2) t_3 2.0) (fma 0.5 (fma t_0 (cos x) t_1) 1.0))
      0.3333333333333333)
     (if (<= x 1.46e-28)
       (/
        (/
         (fma (* (* (pow (sin y) 2.0) -0.0625) (- 1.0 (cos y))) (sqrt 2.0) 2.0)
         (fma 0.5 (fma t_1 (cos y) t_0) 1.0))
        3.0)
       (/
        (/
         (fma t_3 (* t_2 -0.0625) 2.0)
         (fma 1.0 (/ t_1 2.0) (fma (cos x) (/ t_0 2.0) 1.0)))
        3.0)))))

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 = pow(sin(x), 2.0);
	double t_3 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -9.8e-8) {
		tmp = (fma((-0.0625 * t_2), t_3, 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333;
	} else if (x <= 1.46e-28) {
		tmp = (fma(((pow(sin(y), 2.0) * -0.0625) * (1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) / 3.0;
	} else {
		tmp = (fma(t_3, (t_2 * -0.0625), 2.0) / fma(1.0, (t_1 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0))) / 3.0;
	}
	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 = sin(x) ^ 2.0
	t_3 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -9.8e-8)
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_2), t_3, 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333);
	elseif (x <= 1.46e-28)
		tmp = Float64(Float64(fma(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) / 3.0);
	else
		tmp = Float64(Float64(fma(t_3, Float64(t_2 * -0.0625), 2.0) / fma(1.0, Float64(t_1 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0))) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.8e-8], N[(N[(N[(N[(-0.0625 * t$95$2), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.46e-28], N[(N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(t$95$3 * N[(t$95$2 * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.0 * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8000000000000004e-8

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 46.5%

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

   if -9.8000000000000004e-8 < x < 1.46e-28

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.6%

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

   if 1.46e-28 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 99.0%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

   9. Applied rewrites 60.8%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 59.5%

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

Alternative 30: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right) + 0.5 \cdot t\_2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (fma
          (* -0.0625 (pow (sin x) 2.0))
          (* (- (cos x) 1.0) (sqrt 2.0))
          2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -9.8e-8)
     (* (/ t_1 (fma 0.5 (fma t_0 (cos x) t_2) 1.0)) 0.3333333333333333)
     (if (<= x 1.46e-28)
       (/
        (/
         (fma (* (* (pow (sin y) 2.0) -0.0625) (- 1.0 (cos y))) (sqrt 2.0) 2.0)
         (fma 0.5 (fma t_2 (cos y) t_0) 1.0))
        3.0)
       (*
        (/ t_1 (+ (fma (* 0.5 (cos x)) t_0 1.0) (* 0.5 t_2)))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma((-0.0625 * pow(sin(x), 2.0)), ((cos(x) - 1.0) * sqrt(2.0)), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -9.8e-8) {
		tmp = (t_1 / fma(0.5, fma(t_0, cos(x), t_2), 1.0)) * 0.3333333333333333;
	} else if (x <= 1.46e-28) {
		tmp = (fma(((pow(sin(y), 2.0) * -0.0625) * (1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) / 3.0;
	} else {
		tmp = (t_1 / (fma((0.5 * cos(x)), t_0, 1.0) + (0.5 * t_2))) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -9.8e-8)
		tmp = Float64(Float64(t_1 / fma(0.5, fma(t_0, cos(x), t_2), 1.0)) * 0.3333333333333333);
	elseif (x <= 1.46e-28)
		tmp = Float64(Float64(fma(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) / 3.0);
	else
		tmp = Float64(Float64(t_1 / Float64(fma(Float64(0.5 * cos(x)), t_0, 1.0) + Float64(0.5 * t_2))) * 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[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.8e-8], N[(N[(t$95$1 / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.46e-28], N[(N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(t$95$1 / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8000000000000004e-8

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 46.5%

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

   if -9.8000000000000004e-8 < x < 1.46e-28

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.6%

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

   if 1.46e-28 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 59.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lower-+.f64 N/A

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

   7. Applied rewrites 59.4%

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

Alternative 31: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(t\_0, \cos x, t\_1\right)\\ t_3 := {\sin x}^{2}\\ t_4 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_3, t\_4, 2\right)}{\mathsf{fma}\left(0.5, t\_2, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_4, t\_3 \cdot -0.0625, 2\right)}{\mathsf{fma}\left(t\_2, 0.5, 1\right)}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma t_0 (cos x) t_1))
        (t_3 (pow (sin x) 2.0))
        (t_4 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -9.8e-8)
     (* (/ (fma (* -0.0625 t_3) t_4 2.0) (fma 0.5 t_2 1.0)) 0.3333333333333333)
     (if (<= x 1.46e-28)
       (/
        (/
         (fma (* (* (pow (sin y) 2.0) -0.0625) (- 1.0 (cos y))) (sqrt 2.0) 2.0)
         (fma 0.5 (fma t_1 (cos y) t_0) 1.0))
        3.0)
       (/ (/ (fma t_4 (* t_3 -0.0625) 2.0) (fma t_2 0.5 1.0)) 3.0)))))

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(t_0, cos(x), t_1);
	double t_3 = pow(sin(x), 2.0);
	double t_4 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -9.8e-8) {
		tmp = (fma((-0.0625 * t_3), t_4, 2.0) / fma(0.5, t_2, 1.0)) * 0.3333333333333333;
	} else if (x <= 1.46e-28) {
		tmp = (fma(((pow(sin(y), 2.0) * -0.0625) * (1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) / 3.0;
	} else {
		tmp = (fma(t_4, (t_3 * -0.0625), 2.0) / fma(t_2, 0.5, 1.0)) / 3.0;
	}
	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 = fma(t_0, cos(x), t_1)
	t_3 = sin(x) ^ 2.0
	t_4 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -9.8e-8)
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_3), t_4, 2.0) / fma(0.5, t_2, 1.0)) * 0.3333333333333333);
	elseif (x <= 1.46e-28)
		tmp = Float64(Float64(fma(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) / 3.0);
	else
		tmp = Float64(Float64(fma(t_4, Float64(t_3 * -0.0625), 2.0) / fma(t_2, 0.5, 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.8e-8], N[(N[(N[(N[(-0.0625 * t$95$3), $MachinePrecision] * t$95$4 + 2.0), $MachinePrecision] / N[(0.5 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.46e-28], N[(N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(t$95$4 * N[(t$95$3 * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$2 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8000000000000004e-8

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 46.5%

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

   if -9.8000000000000004e-8 < x < 1.46e-28

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.6%

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

   if 1.46e-28 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 99.0%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 59.4%

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

Alternative 32: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(t\_0, \cos x, t\_1\right)\\ t_3 := {\sin x}^{2}\\ t_4 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_3, t\_4, 2\right)}{\mathsf{fma}\left(0.5, t\_2, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_4, t\_3 \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(t\_2, 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma t_0 (cos x) t_1))
        (t_3 (pow (sin x) 2.0))
        (t_4 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -9.8e-8)
     (* (/ (fma (* -0.0625 t_3) t_4 2.0) (fma 0.5 t_2 1.0)) 0.3333333333333333)
     (if (<= x 1.46e-28)
       (/
        (/
         (fma (* (* (pow (sin y) 2.0) -0.0625) (- 1.0 (cos y))) (sqrt 2.0) 2.0)
         (fma 0.5 (fma t_1 (cos y) t_0) 1.0))
        3.0)
       (/
        (* (fma t_4 (* t_3 -0.0625) 2.0) 0.3333333333333333)
        (fma t_2 0.5 1.0))))))

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(t_0, cos(x), t_1);
	double t_3 = pow(sin(x), 2.0);
	double t_4 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -9.8e-8) {
		tmp = (fma((-0.0625 * t_3), t_4, 2.0) / fma(0.5, t_2, 1.0)) * 0.3333333333333333;
	} else if (x <= 1.46e-28) {
		tmp = (fma(((pow(sin(y), 2.0) * -0.0625) * (1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) / 3.0;
	} else {
		tmp = (fma(t_4, (t_3 * -0.0625), 2.0) * 0.3333333333333333) / fma(t_2, 0.5, 1.0);
	}
	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 = fma(t_0, cos(x), t_1)
	t_3 = sin(x) ^ 2.0
	t_4 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -9.8e-8)
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_3), t_4, 2.0) / fma(0.5, t_2, 1.0)) * 0.3333333333333333);
	elseif (x <= 1.46e-28)
		tmp = Float64(Float64(fma(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(1.0 - cos(y))), sqrt(2.0), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) / 3.0);
	else
		tmp = Float64(Float64(fma(t_4, Float64(t_3 * -0.0625), 2.0) * 0.3333333333333333) / fma(t_2, 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.8e-8], N[(N[(N[(N[(-0.0625 * t$95$3), $MachinePrecision] * t$95$4 + 2.0), $MachinePrecision] / N[(0.5 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.46e-28], N[(N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(t$95$4 * N[(t$95$3 * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(t$95$2 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8000000000000004e-8

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 46.5%

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

   if -9.8000000000000004e-8 < x < 1.46e-28

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.6%

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

   if 1.46e-28 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 59.4%

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

   7. Applied rewrites 59.4%

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

Alternative 33: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(t\_0, \cos x, t\_1\right)\\ t_3 := {\sin x}^{2}\\ t_4 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_3, t\_4, 2\right)}{\mathsf{fma}\left(0.5, t\_2, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_4, t\_3 \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(t\_2, 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma t_0 (cos x) t_1))
        (t_3 (pow (sin x) 2.0))
        (t_4 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -9.8e-8)
     (* (/ (fma (* -0.0625 t_3) t_4 2.0) (fma 0.5 t_2 1.0)) 0.3333333333333333)
     (if (<= x 1.46e-28)
       (*
        (/
         (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
         (fma 0.5 (fma t_1 (cos y) t_0) 1.0))
        0.3333333333333333)
       (/
        (* (fma t_4 (* t_3 -0.0625) 2.0) 0.3333333333333333)
        (fma t_2 0.5 1.0))))))

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(t_0, cos(x), t_1);
	double t_3 = pow(sin(x), 2.0);
	double t_4 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -9.8e-8) {
		tmp = (fma((-0.0625 * t_3), t_4, 2.0) / fma(0.5, t_2, 1.0)) * 0.3333333333333333;
	} else if (x <= 1.46e-28) {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (fma(t_4, (t_3 * -0.0625), 2.0) * 0.3333333333333333) / fma(t_2, 0.5, 1.0);
	}
	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 = fma(t_0, cos(x), t_1)
	t_3 = sin(x) ^ 2.0
	t_4 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -9.8e-8)
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_3), t_4, 2.0) / fma(0.5, t_2, 1.0)) * 0.3333333333333333);
	elseif (x <= 1.46e-28)
		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_1, cos(y), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(t_4, Float64(t_3 * -0.0625), 2.0) * 0.3333333333333333) / fma(t_2, 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.8e-8], N[(N[(N[(N[(-0.0625 * t$95$3), $MachinePrecision] * t$95$4 + 2.0), $MachinePrecision] / N[(0.5 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.46e-28], 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$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(t$95$4 * N[(t$95$3 * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(t$95$2 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8000000000000004e-8

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 46.5%

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

   if -9.8000000000000004e-8 < x < 1.46e-28

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.4%

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

   if 1.46e-28 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 59.4%

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

   7. Applied rewrites 59.4%

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

Alternative 34: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right)\\ t_3 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, t\_3, 2\right)}{t\_2} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_3, 2\right)}{t\_2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma 0.5 (fma t_0 (cos x) t_1) 1.0))
        (t_3 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -9.8e-8)
     (* (/ (fma (* -0.0625 (pow (sin x) 2.0)) t_3 2.0) t_2) 0.3333333333333333)
     (if (<= x 1.46e-28)
       (*
        (/
         (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
         (fma 0.5 (fma t_1 (cos y) t_0) 1.0))
        0.3333333333333333)
       (*
        (/ (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_3 2.0) t_2)
        0.3333333333333333)))))

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(0.5, fma(t_0, cos(x), t_1), 1.0);
	double t_3 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -9.8e-8) {
		tmp = (fma((-0.0625 * pow(sin(x), 2.0)), t_3, 2.0) / t_2) * 0.3333333333333333;
	} else if (x <= 1.46e-28) {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), t_3, 2.0) / t_2) * 0.3333333333333333;
	}
	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 = fma(0.5, fma(t_0, cos(x), t_1), 1.0)
	t_3 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -9.8e-8)
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(x) ^ 2.0)), t_3, 2.0) / t_2) * 0.3333333333333333);
	elseif (x <= 1.46e-28)
		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_1, cos(y), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), t_3, 2.0) / t_2) * 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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.8e-8], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.46e-28], 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$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], 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$3 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8000000000000004e-8

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 46.5%

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

   if -9.8000000000000004e-8 < x < 1.46e-28

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 99.4%

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

   if 1.46e-28 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 59.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 59.4

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

   7. Applied rewrites 59.4%

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

Alternative 35: 60.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma (* -0.0625 (pow (sin x) 2.0)) (* (- (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 * pow(sin(x), 2.0)), ((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 * (sin(x) ^ 2.0)), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 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[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $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[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 99.2%

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

3. Taylor expanded in y around 0

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

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

5. Applied rewrites 58.3%

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

Alternative 36: 60.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma (* -0.0625 (pow (sin x) 2.0)) (* (- (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 * pow(sin(x), 2.0)), ((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 * (sin(x) ^ 2.0)), 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[Power[N[Sin[x], $MachinePrecision], 2.0], $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 99.2%

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

3. Taylor expanded in y around 0

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

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

5. Applied rewrites 58.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. associate-+r- N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. lower--.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lift-cos.f64 58.3

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

7. Applied rewrites 58.3%

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

Alternative 37: 60.2% accurate, 2.4× speedup

Math

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

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, 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[(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[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 99.2%

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

3. Taylor expanded in y around 0

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

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

5. Applied rewrites 58.3%

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

   1. lift-pow.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. unpow2 N/A

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

   4. sqr-sin-a N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 57.9

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

7. Applied rewrites 57.9%

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

Alternative 38: 45.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

7. Applied rewrites 59.9%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 46.0%

    \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right)} \]
11. Add Preprocessing

Alternative 39: 43.4% accurate, 6.1× 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 99.2%

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

3. Taylor expanded in y around 0

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

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

5. Applied rewrites 58.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 43.7%

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

Alternative 40: 40.9% accurate, 940.0× 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 99.2%

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

3. Taylor expanded in y around 0

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

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

5. Applied rewrites 58.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
7. Step-by-step derivation

8. Applied rewrites 41.4%

   \[\leadsto 0.3333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025043 
(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))))))