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

Percentage Accurate: 99.3% → 99.3%

Time: 10.4s

Alternatives: 34

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-neg-out N/A

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

   9. lift-*.f64 N/A

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

5. Applied rewrites 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval 99.3

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

5. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval 99.3

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

5. Applied rewrites 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), -1.5, -3\right)}} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

Alternative 5: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3\\ t_1 := \cos y - \cos x\\ t_2 := \frac{\mathsf{fma}\left(t\_1, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\left(\sin x + y \cdot \left(0.010416666666666666 \cdot {y}^{2} - 0.0625\right)\right) \cdot \sqrt{2}\right), -2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (-
          (*
           (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
           -1.5)
          3.0))
        (t_1 (- (cos y) (cos x)))
        (t_2
         (/
          (fma
           t_1
           (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
           -2.0)
          t_0)))
   (if (<= y -0.15)
     t_2
     (if (<= y 0.175)
       (/
        (fma
         t_1
         (*
          (fma (sin x) -0.0625 (sin y))
          (*
           (+ (sin x) (* y (- (* 0.010416666666666666 (pow y 2.0)) 0.0625)))
           (sqrt 2.0)))
         -2.0)
        t_0)
       t_2))))

C

double code(double x, double y) {
	double t_0 = (fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))) * -1.5) - 3.0;
	double t_1 = cos(y) - cos(x);
	double t_2 = fma(t_1, (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / t_0;
	double tmp;
	if (y <= -0.15) {
		tmp = t_2;
	} else if (y <= 0.175) {
		tmp = fma(t_1, (fma(sin(x), -0.0625, sin(y)) * ((sin(x) + (y * ((0.010416666666666666 * pow(y, 2.0)) - 0.0625))) * sqrt(2.0))), -2.0) / t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) * -1.5) - 3.0)
	t_1 = Float64(cos(y) - cos(x))
	t_2 = Float64(fma(t_1, Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / t_0)
	tmp = 0.0
	if (y <= -0.15)
		tmp = t_2;
	elseif (y <= 0.175)
		tmp = Float64(fma(t_1, Float64(fma(sin(x), -0.0625, sin(y)) * Float64(Float64(sin(x) + Float64(y * Float64(Float64(0.010416666666666666 * (y ^ 2.0)) - 0.0625))) * sqrt(2.0))), -2.0) / t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.15], t$95$2, If[LessEqual[y, 0.175], N[(N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(y * N[(N[(0.010416666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.149999999999999994 or 0.17499999999999999 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.7

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

   8. Applied rewrites 64.7%

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

   if -0.149999999999999994 < y < 0.17499999999999999

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 49.5

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\left(\sin x + y \cdot \left(0.010416666666666666 \cdot {y}^{2} - 0.0625\right)\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3} \]

   8. Applied rewrites 49.5%

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

Alternative 6: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)\\ t_1 := \cos y - \cos x\\ t_2 := \frac{\mathsf{fma}\left(t\_1, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{t\_0 \cdot -1.5 - 3}\\ \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\left(\sin x + y \cdot \left(0.010416666666666666 \cdot {y}^{2} - 0.0625\right)\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(t\_0, -1.5, -3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))))
        (t_1 (- (cos y) (cos x)))
        (t_2
         (/
          (fma
           t_1
           (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
           -2.0)
          (- (* t_0 -1.5) 3.0))))
   (if (<= y -0.15)
     t_2
     (if (<= y 0.175)
       (/
        (fma
         t_1
         (*
          (fma (sin x) -0.0625 (sin y))
          (*
           (+ (sin x) (* y (- (* 0.010416666666666666 (pow y 2.0)) 0.0625)))
           (sqrt 2.0)))
         -2.0)
        (fma t_0 -1.5 -3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y)));
	double t_1 = cos(y) - cos(x);
	double t_2 = fma(t_1, (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / ((t_0 * -1.5) - 3.0);
	double tmp;
	if (y <= -0.15) {
		tmp = t_2;
	} else if (y <= 0.175) {
		tmp = fma(t_1, (fma(sin(x), -0.0625, sin(y)) * ((sin(x) + (y * ((0.010416666666666666 * pow(y, 2.0)) - 0.0625))) * sqrt(2.0))), -2.0) / fma(t_0, -1.5, -3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y)))
	t_1 = Float64(cos(y) - cos(x))
	t_2 = Float64(fma(t_1, Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / Float64(Float64(t_0 * -1.5) - 3.0))
	tmp = 0.0
	if (y <= -0.15)
		tmp = t_2;
	elseif (y <= 0.175)
		tmp = Float64(fma(t_1, Float64(fma(sin(x), -0.0625, sin(y)) * Float64(Float64(sin(x) + Float64(y * Float64(Float64(0.010416666666666666 * (y ^ 2.0)) - 0.0625))) * sqrt(2.0))), -2.0) / fma(t_0, -1.5, -3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(t$95$0 * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.15], t$95$2, If[LessEqual[y, 0.175], N[(N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(y * N[(N[(0.010416666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(t$95$0 * -1.5 + -3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.149999999999999994 or 0.17499999999999999 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.7

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

   8. Applied rewrites 64.7%

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

   if -0.149999999999999994 < y < 0.17499999999999999

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 49.5

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\left(\sin x + y \cdot \left(0.010416666666666666 \cdot {y}^{2} - 0.0625\right)\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), -1.5, -3\right)} \]

   8. Applied rewrites 49.5%

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

Alternative 7: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3\\ t_1 := y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\\ t_2 := \cos y - \cos x\\ t_3 := \frac{\mathsf{fma}\left(t\_2, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.13:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 0.078:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \mathsf{fma}\left(\sin x, -0.0625, t\_1\right) \cdot \left(\mathsf{fma}\left(t\_1, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (-
          (*
           (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
           -1.5)
          3.0))
        (t_1 (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0)))))
        (t_2 (- (cos y) (cos x)))
        (t_3
         (/
          (fma
           t_2
           (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
           -2.0)
          t_0)))
   (if (<= y -0.13)
     t_3
     (if (<= y 0.078)
       (/
        (fma
         t_2
         (* (fma (sin x) -0.0625 t_1) (* (fma t_1 -0.0625 (sin x)) (sqrt 2.0)))
         -2.0)
        t_0)
       t_3))))

C

double code(double x, double y) {
	double t_0 = (fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))) * -1.5) - 3.0;
	double t_1 = y * (1.0 + (-0.16666666666666666 * pow(y, 2.0)));
	double t_2 = cos(y) - cos(x);
	double t_3 = fma(t_2, (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / t_0;
	double tmp;
	if (y <= -0.13) {
		tmp = t_3;
	} else if (y <= 0.078) {
		tmp = fma(t_2, (fma(sin(x), -0.0625, t_1) * (fma(t_1, -0.0625, sin(x)) * sqrt(2.0))), -2.0) / t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) * -1.5) - 3.0)
	t_1 = Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0))))
	t_2 = Float64(cos(y) - cos(x))
	t_3 = Float64(fma(t_2, Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / t_0)
	tmp = 0.0
	if (y <= -0.13)
		tmp = t_3;
	elseif (y <= 0.078)
		tmp = Float64(fma(t_2, Float64(fma(sin(x), -0.0625, t_1) * Float64(fma(t_1, -0.0625, sin(x)) * sqrt(2.0))), -2.0) / t_0);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.13], t$95$3, If[LessEqual[y, 0.078], N[(N[(t$95$2 * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + t$95$1), $MachinePrecision] * N[(N[(t$95$1 * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.13 or 0.0779999999999999999 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.7

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

   8. Applied rewrites 64.7%

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

   if -0.13 < y < 0.0779999999999999999

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 49.5

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{\color{blue}{2}}\right)\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3} \]

   8. Applied rewrites 49.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \color{blue}{y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3} \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 49.0

          \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{\color{blue}{2}}\right), -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3} \]

   11. Applied rewrites 49.0%

       \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\\ t_3 := \frac{\mathsf{fma}\left(\cos y - \cos x, \sin y \cdot t\_2, -2\right)}{\mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right) \cdot -1.5 - 3}\\ \mathbf{if}\;y \leq -0.09:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 0.07:\\ \;\;\;\;\left(2 - \left(\left(\mathsf{fma}\left(y \cdot y, -0.5, 1\right) - \cos x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot t\_2\right) \cdot \frac{0.3333333333333333}{1 + 0.5 \cdot \mathsf{fma}\left(\cos x, t\_0, \cos y \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
        (t_3
         (/
          (fma (- (cos y) (cos x)) (* (sin y) t_2) -2.0)
          (- (* (fma t_0 (cos x) (* t_1 (cos y))) -1.5) 3.0))))
   (if (<= y -0.09)
     t_3
     (if (<= y 0.07)
       (*
        (-
         2.0
         (*
          (* (- (fma (* y y) -0.5 1.0) (cos x)) (fma -0.0625 (sin x) (sin y)))
          t_2))
        (/
         0.3333333333333333
         (+ 1.0 (* 0.5 (fma (cos x) t_0 (* (cos y) t_1))))))
       t_3))))

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(sin(y), -0.0625, sin(x)) * sqrt(2.0);
	double t_3 = fma((cos(y) - cos(x)), (sin(y) * t_2), -2.0) / ((fma(t_0, cos(x), (t_1 * cos(y))) * -1.5) - 3.0);
	double tmp;
	if (y <= -0.09) {
		tmp = t_3;
	} else if (y <= 0.07) {
		tmp = (2.0 - (((fma((y * y), -0.5, 1.0) - cos(x)) * fma(-0.0625, sin(x), sin(y))) * t_2)) * (0.3333333333333333 / (1.0 + (0.5 * fma(cos(x), t_0, (cos(y) * t_1)))));
	} else {
		tmp = t_3;
	}
	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(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))
	t_3 = Float64(fma(Float64(cos(y) - cos(x)), Float64(sin(y) * t_2), -2.0) / Float64(Float64(fma(t_0, cos(x), Float64(t_1 * cos(y))) * -1.5) - 3.0))
	tmp = 0.0
	if (y <= -0.09)
		tmp = t_3;
	elseif (y <= 0.07)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(fma(Float64(y * y), -0.5, 1.0) - cos(x)) * fma(-0.0625, sin(x), sin(y))) * t_2)) * Float64(0.3333333333333333 / Float64(1.0 + Float64(0.5 * fma(cos(x), t_0, Float64(cos(y) * t_1))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$2), $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.09], t$95$3, If[LessEqual[y, 0.07], N[(N[(2.0 - N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.089999999999999997 or 0.070000000000000007 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.7

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

   8. Applied rewrites 64.7%

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

   if -0.089999999999999997 < y < 0.070000000000000007

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 49.1

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.6

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

   9. Applied rewrites 51.6%

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

   11. Applied rewrites 51.6%

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

   12. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-cos.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-sqrt.f64 49.1

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

   14. Applied rewrites 49.1%

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

Alternative 9: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_4 := \frac{\mathsf{fma}\left(\cos y - \cos x, \sin y \cdot \left(t\_3 \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(t\_0, \cos x, t\_2 \cdot \cos y\right) \cdot -1.5 - 3}\\ \mathbf{if}\;y \leq -0.0135:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;y \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_3\right) \cdot \sqrt{2}, t\_1 - \cos x, -2\right) \cdot -0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot t\_2\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (fma (* y y) -0.5 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma (sin y) -0.0625 (sin x)))
        (t_4
         (/
          (fma (- (cos y) (cos x)) (* (sin y) (* t_3 (sqrt 2.0))) -2.0)
          (- (* (fma t_0 (cos x) (* t_2 (cos y))) -1.5) 3.0))))
   (if (<= y -0.0135)
     t_4
     (if (<= y 0.02)
       (/
        (*
         (fma
          (* (* (fma -0.0625 (sin x) (sin y)) t_3) (sqrt 2.0))
          (- t_1 (cos x))
          -2.0)
         -0.3333333333333333)
        (fma (fma t_0 (cos x) (* t_1 t_2)) 0.5 1.0))
       t_4))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma((y * y), -0.5, 1.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(sin(y), -0.0625, sin(x));
	double t_4 = fma((cos(y) - cos(x)), (sin(y) * (t_3 * sqrt(2.0))), -2.0) / ((fma(t_0, cos(x), (t_2 * cos(y))) * -1.5) - 3.0);
	double tmp;
	if (y <= -0.0135) {
		tmp = t_4;
	} else if (y <= 0.02) {
		tmp = (fma(((fma(-0.0625, sin(x), sin(y)) * t_3) * sqrt(2.0)), (t_1 - cos(x)), -2.0) * -0.3333333333333333) / fma(fma(t_0, cos(x), (t_1 * t_2)), 0.5, 1.0);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(Float64(y * y), -0.5, 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(sin(y), -0.0625, sin(x))
	t_4 = Float64(fma(Float64(cos(y) - cos(x)), Float64(sin(y) * Float64(t_3 * sqrt(2.0))), -2.0) / Float64(Float64(fma(t_0, cos(x), Float64(t_2 * cos(y))) * -1.5) - 3.0))
	tmp = 0.0
	if (y <= -0.0135)
		tmp = t_4;
	elseif (y <= 0.02)
		tmp = Float64(Float64(fma(Float64(Float64(fma(-0.0625, sin(x), sin(y)) * t_3) * sqrt(2.0)), Float64(t_1 - cos(x)), -2.0) * -0.3333333333333333) / fma(fma(t_0, cos(x), Float64(t_1 * t_2)), 0.5, 1.0));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0135], t$95$4, If[LessEqual[y, 0.02], N[(N[(N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0134999999999999998 or 0.0200000000000000004 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.7

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

   8. Applied rewrites 64.7%

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

   if -0.0134999999999999998 < y < 0.0200000000000000004

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 49.1

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.6

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

   9. Applied rewrites 51.6%

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

   11. Applied rewrites 51.6%

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

Alternative 10: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_4 := \frac{\mathsf{fma}\left(\cos y - \cos x, \sin y \cdot \left(t\_3 \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_2 \cdot \cos y\right), -1.5, -3\right)}\\ \mathbf{if}\;y \leq -0.0135:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;y \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_3\right) \cdot \sqrt{2}, t\_1 - \cos x, -2\right) \cdot -0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot t\_2\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (fma (* y y) -0.5 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma (sin y) -0.0625 (sin x)))
        (t_4
         (/
          (fma (- (cos y) (cos x)) (* (sin y) (* t_3 (sqrt 2.0))) -2.0)
          (fma (fma t_0 (cos x) (* t_2 (cos y))) -1.5 -3.0))))
   (if (<= y -0.0135)
     t_4
     (if (<= y 0.02)
       (/
        (*
         (fma
          (* (* (fma -0.0625 (sin x) (sin y)) t_3) (sqrt 2.0))
          (- t_1 (cos x))
          -2.0)
         -0.3333333333333333)
        (fma (fma t_0 (cos x) (* t_1 t_2)) 0.5 1.0))
       t_4))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma((y * y), -0.5, 1.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(sin(y), -0.0625, sin(x));
	double t_4 = fma((cos(y) - cos(x)), (sin(y) * (t_3 * sqrt(2.0))), -2.0) / fma(fma(t_0, cos(x), (t_2 * cos(y))), -1.5, -3.0);
	double tmp;
	if (y <= -0.0135) {
		tmp = t_4;
	} else if (y <= 0.02) {
		tmp = (fma(((fma(-0.0625, sin(x), sin(y)) * t_3) * sqrt(2.0)), (t_1 - cos(x)), -2.0) * -0.3333333333333333) / fma(fma(t_0, cos(x), (t_1 * t_2)), 0.5, 1.0);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(Float64(y * y), -0.5, 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(sin(y), -0.0625, sin(x))
	t_4 = Float64(fma(Float64(cos(y) - cos(x)), Float64(sin(y) * Float64(t_3 * sqrt(2.0))), -2.0) / fma(fma(t_0, cos(x), Float64(t_2 * cos(y))), -1.5, -3.0))
	tmp = 0.0
	if (y <= -0.0135)
		tmp = t_4;
	elseif (y <= 0.02)
		tmp = Float64(Float64(fma(Float64(Float64(fma(-0.0625, sin(x), sin(y)) * t_3) * sqrt(2.0)), Float64(t_1 - cos(x)), -2.0) * -0.3333333333333333) / fma(fma(t_0, cos(x), Float64(t_1 * t_2)), 0.5, 1.0));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0135], t$95$4, If[LessEqual[y, 0.02], N[(N[(N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0134999999999999998 or 0.0200000000000000004 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.7

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

   8. Applied rewrites 64.7%

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

   if -0.0134999999999999998 < y < 0.0200000000000000004

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 49.1

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.6

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

   9. Applied rewrites 51.6%

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

   11. Applied rewrites 51.6%

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

Alternative 11: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_3 := 3 - \sqrt{5}\\ t_4 := t\_3 \cdot \cos y\\ t_5 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ \mathbf{if}\;y \leq -0.0135:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right) - 2}{\mathsf{fma}\left(t\_1, \cos x, t\_4\right) \cdot -1.5 - 3}\\ \mathbf{elif}\;y \leq 0.043:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_2\right) \cdot \sqrt{2}, t\_5 - \cos x, -2\right) \cdot -0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_5 \cdot t\_3\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(t\_0, \sin y \cdot \left(t\_2 \cdot \sqrt{2}\right), -2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, 1, t\_4\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos y) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (fma (sin y) -0.0625 (sin x)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (* t_3 (cos y)))
        (t_5 (fma (* y y) -0.5 1.0)))
   (if (<= y -0.0135)
     (/
      (- (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) t_0))) 2.0)
      (- (* (fma t_1 (cos x) t_4) -1.5) 3.0))
     (if (<= y 0.043)
       (/
        (*
         (fma
          (* (* (fma -0.0625 (sin x) (sin y)) t_2) (sqrt 2.0))
          (- t_5 (cos x))
          -2.0)
         -0.3333333333333333)
        (fma (fma t_1 (cos x) (* t_5 t_3)) 0.5 1.0))
       (-
        (/
         (fma t_0 (* (sin y) (* t_2 (sqrt 2.0))) -2.0)
         (+ 3.0 (* 3.0 (/ (fma t_1 1.0 t_4) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(y) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma(sin(y), -0.0625, sin(x));
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 * cos(y);
	double t_5 = fma((y * y), -0.5, 1.0);
	double tmp;
	if (y <= -0.0135) {
		tmp = ((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * t_0))) - 2.0) / ((fma(t_1, cos(x), t_4) * -1.5) - 3.0);
	} else if (y <= 0.043) {
		tmp = (fma(((fma(-0.0625, sin(x), sin(y)) * t_2) * sqrt(2.0)), (t_5 - cos(x)), -2.0) * -0.3333333333333333) / fma(fma(t_1, cos(x), (t_5 * t_3)), 0.5, 1.0);
	} else {
		tmp = -(fma(t_0, (sin(y) * (t_2 * sqrt(2.0))), -2.0) / (3.0 + (3.0 * (fma(t_1, 1.0, t_4) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(sin(y), -0.0625, sin(x))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 * cos(y))
	t_5 = fma(Float64(y * y), -0.5, 1.0)
	tmp = 0.0
	if (y <= -0.0135)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * t_0))) - 2.0) / Float64(Float64(fma(t_1, cos(x), t_4) * -1.5) - 3.0));
	elseif (y <= 0.043)
		tmp = Float64(Float64(fma(Float64(Float64(fma(-0.0625, sin(x), sin(y)) * t_2) * sqrt(2.0)), Float64(t_5 - cos(x)), -2.0) * -0.3333333333333333) / fma(fma(t_1, cos(x), Float64(t_5 * t_3)), 0.5, 1.0));
	else
		tmp = Float64(-Float64(fma(t_0, Float64(sin(y) * Float64(t_2 * sqrt(2.0))), -2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, 1.0, t_4) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[y, -0.0135], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$4), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.043], N[(N[(N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$5 - N[Cos[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$5 * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], (-N[(N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * 1.0 + t$95$4), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0134999999999999998

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if -0.0134999999999999998 < y < 0.042999999999999997

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 49.1

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.6

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

   9. Applied rewrites 51.6%

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

   11. Applied rewrites 51.6%

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

   if 0.042999999999999997 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.7%

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

   10. Taylor expanded in x around 0

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

       1. lower-sin.f64 60.6

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

   12. Applied rewrites 60.6%

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

Alternative 12: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y - 1\\ t_1 := 1 + -0.5 \cdot {y}^{2}\\ t_2 := \sqrt{5} - 1\\ t_3 := 3 - \sqrt{5}\\ t_4 := t\_3 \cdot \cos y\\ \mathbf{if}\;y \leq -0.0135:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right) - 2}{\mathsf{fma}\left(t\_2, \cos x, t\_4\right) \cdot -1.5 - 3}\\ \mathbf{elif}\;y \leq 14:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\left(\sin x + -0.0625 \cdot y\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(t\_2, \cos x, t\_3 \cdot t\_1\right)}{2} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(t\_0, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_2, 1, t\_4\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos y) 1.0))
        (t_1 (+ 1.0 (* -0.5 (pow y 2.0))))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (* t_3 (cos y))))
   (if (<= y -0.0135)
     (/
      (- (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) t_0))) 2.0)
      (- (* (fma t_2 (cos x) t_4) -1.5) 3.0))
     (if (<= y 14.0)
       (/
        (fma
         (- t_1 (cos x))
         (*
          (fma (sin x) -0.0625 (sin y))
          (* (+ (sin x) (* -0.0625 y)) (sqrt 2.0)))
         -2.0)
        (* -3.0 (+ (/ (fma t_2 (cos x) (* t_3 t_1)) 2.0) 1.0)))
       (-
        (/
         (fma
          t_0
          (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
          -2.0)
         (+ 3.0 (* 3.0 (/ (fma t_2 1.0 t_4) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(y) - 1.0;
	double t_1 = 1.0 + (-0.5 * pow(y, 2.0));
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 * cos(y);
	double tmp;
	if (y <= -0.0135) {
		tmp = ((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * t_0))) - 2.0) / ((fma(t_2, cos(x), t_4) * -1.5) - 3.0);
	} else if (y <= 14.0) {
		tmp = fma((t_1 - cos(x)), (fma(sin(x), -0.0625, sin(y)) * ((sin(x) + (-0.0625 * y)) * sqrt(2.0))), -2.0) / (-3.0 * ((fma(t_2, cos(x), (t_3 * t_1)) / 2.0) + 1.0));
	} else {
		tmp = -(fma(t_0, (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / (3.0 + (3.0 * (fma(t_2, 1.0, t_4) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) - 1.0)
	t_1 = Float64(1.0 + Float64(-0.5 * (y ^ 2.0)))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 * cos(y))
	tmp = 0.0
	if (y <= -0.0135)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * t_0))) - 2.0) / Float64(Float64(fma(t_2, cos(x), t_4) * -1.5) - 3.0));
	elseif (y <= 14.0)
		tmp = Float64(fma(Float64(t_1 - cos(x)), Float64(fma(sin(x), -0.0625, sin(y)) * Float64(Float64(sin(x) + Float64(-0.0625 * y)) * sqrt(2.0))), -2.0) / Float64(-3.0 * Float64(Float64(fma(t_2, cos(x), Float64(t_3 * t_1)) / 2.0) + 1.0)));
	else
		tmp = Float64(-Float64(fma(t_0, Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_2, 1.0, t_4) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0135], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$4), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 14.0], N[(N[(N[(t$95$1 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(-3.0 * N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$2 * 1.0 + t$95$4), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0134999999999999998

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if -0.0134999999999999998 < y < 14

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 49.1

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.6

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

   9. Applied rewrites 51.6%

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

   10. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 49.1

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

   12. Applied rewrites 49.1%

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

   if 14 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.7%

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

   10. Taylor expanded in x around 0

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

       1. lower-sin.f64 60.6

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

   12. Applied rewrites 60.6%

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

Alternative 13: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_3 := 3 - \sqrt{5}\\ t_4 := t\_3 \cdot \cos y\\ \mathbf{if}\;y \leq -0.0135:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right) - 2}{\mathsf{fma}\left(t\_1, \cos x, t\_4\right) \cdot -1.5 - 3}\\ \mathbf{elif}\;y \leq 0.043:\\ \;\;\;\;\left(2 - \left(\left(t\_2 - \cos x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot \left(\left(\sin x + y \cdot \left(0.010416666666666666 \cdot {y}^{2} - 0.0625\right)\right) \cdot \sqrt{2}\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_2 \cdot t\_3\right), 0.5, 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(t\_0, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, 1, t\_4\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos y) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (fma (* y y) -0.5 1.0))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (* t_3 (cos y))))
   (if (<= y -0.0135)
     (/
      (- (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) t_0))) 2.0)
      (- (* (fma t_1 (cos x) t_4) -1.5) 3.0))
     (if (<= y 0.043)
       (*
        (-
         2.0
         (*
          (* (- t_2 (cos x)) (fma -0.0625 (sin x) (sin y)))
          (*
           (+ (sin x) (* y (- (* 0.010416666666666666 (pow y 2.0)) 0.0625)))
           (sqrt 2.0))))
        (/ 1.0 (* (fma (fma t_1 (cos x) (* t_2 t_3)) 0.5 1.0) 3.0)))
       (-
        (/
         (fma
          t_0
          (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
          -2.0)
         (+ 3.0 (* 3.0 (/ (fma t_1 1.0 t_4) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(y) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma((y * y), -0.5, 1.0);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 * cos(y);
	double tmp;
	if (y <= -0.0135) {
		tmp = ((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * t_0))) - 2.0) / ((fma(t_1, cos(x), t_4) * -1.5) - 3.0);
	} else if (y <= 0.043) {
		tmp = (2.0 - (((t_2 - cos(x)) * fma(-0.0625, sin(x), sin(y))) * ((sin(x) + (y * ((0.010416666666666666 * pow(y, 2.0)) - 0.0625))) * sqrt(2.0)))) * (1.0 / (fma(fma(t_1, cos(x), (t_2 * t_3)), 0.5, 1.0) * 3.0));
	} else {
		tmp = -(fma(t_0, (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / (3.0 + (3.0 * (fma(t_1, 1.0, t_4) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(Float64(y * y), -0.5, 1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 * cos(y))
	tmp = 0.0
	if (y <= -0.0135)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * t_0))) - 2.0) / Float64(Float64(fma(t_1, cos(x), t_4) * -1.5) - 3.0));
	elseif (y <= 0.043)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(t_2 - cos(x)) * fma(-0.0625, sin(x), sin(y))) * Float64(Float64(sin(x) + Float64(y * Float64(Float64(0.010416666666666666 * (y ^ 2.0)) - 0.0625))) * sqrt(2.0)))) * Float64(1.0 / Float64(fma(fma(t_1, cos(x), Float64(t_2 * t_3)), 0.5, 1.0) * 3.0)));
	else
		tmp = Float64(-Float64(fma(t_0, Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, 1.0, t_4) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0135], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$4), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.043], N[(N[(2.0 - N[(N[(N[(t$95$2 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(y * N[(N[(0.010416666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * 1.0 + t$95$4), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0134999999999999998

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if -0.0134999999999999998 < y < 0.042999999999999997

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 49.1

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.6

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

   9. Applied rewrites 51.6%

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

   11. Applied rewrites 51.6%

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

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-pow.f64 48.9

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

   14. Applied rewrites 48.9%

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

   if 0.042999999999999997 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.7%

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

   10. Taylor expanded in x around 0

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

       1. lower-sin.f64 60.6

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

   12. Applied rewrites 60.6%

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

Alternative 14: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_3 := 3 - \sqrt{5}\\ t_4 := t\_3 \cdot \cos y\\ t_5 := y \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right)\\ \mathbf{if}\;y \leq -0.0135:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right) - 2}{\mathsf{fma}\left(t\_1, \cos x, t\_4\right) \cdot -1.5 - 3}\\ \mathbf{elif}\;y \leq 0.043:\\ \;\;\;\;\left(2 - \left(\left(t\_2 - \cos x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, t\_5\right)\right) \cdot \left(\mathsf{fma}\left(t\_5, -0.0625, \sin x\right) \cdot \sqrt{2}\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_2 \cdot t\_3\right), 0.5, 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(t\_0, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, 1, t\_4\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos y) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (fma (* y y) -0.5 1.0))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (* t_3 (cos y)))
        (t_5 (* y (+ 1.0 (* -0.16666666666666666 (pow y 2.0))))))
   (if (<= y -0.0135)
     (/
      (- (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) t_0))) 2.0)
      (- (* (fma t_1 (cos x) t_4) -1.5) 3.0))
     (if (<= y 0.043)
       (*
        (-
         2.0
         (*
          (* (- t_2 (cos x)) (fma -0.0625 (sin x) t_5))
          (* (fma t_5 -0.0625 (sin x)) (sqrt 2.0))))
        (/ 1.0 (* (fma (fma t_1 (cos x) (* t_2 t_3)) 0.5 1.0) 3.0)))
       (-
        (/
         (fma
          t_0
          (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
          -2.0)
         (+ 3.0 (* 3.0 (/ (fma t_1 1.0 t_4) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(y) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma((y * y), -0.5, 1.0);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 * cos(y);
	double t_5 = y * (1.0 + (-0.16666666666666666 * pow(y, 2.0)));
	double tmp;
	if (y <= -0.0135) {
		tmp = ((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * t_0))) - 2.0) / ((fma(t_1, cos(x), t_4) * -1.5) - 3.0);
	} else if (y <= 0.043) {
		tmp = (2.0 - (((t_2 - cos(x)) * fma(-0.0625, sin(x), t_5)) * (fma(t_5, -0.0625, sin(x)) * sqrt(2.0)))) * (1.0 / (fma(fma(t_1, cos(x), (t_2 * t_3)), 0.5, 1.0) * 3.0));
	} else {
		tmp = -(fma(t_0, (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / (3.0 + (3.0 * (fma(t_1, 1.0, t_4) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(Float64(y * y), -0.5, 1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 * cos(y))
	t_5 = Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (y ^ 2.0))))
	tmp = 0.0
	if (y <= -0.0135)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * t_0))) - 2.0) / Float64(Float64(fma(t_1, cos(x), t_4) * -1.5) - 3.0));
	elseif (y <= 0.043)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(t_2 - cos(x)) * fma(-0.0625, sin(x), t_5)) * Float64(fma(t_5, -0.0625, sin(x)) * sqrt(2.0)))) * Float64(1.0 / Float64(fma(fma(t_1, cos(x), Float64(t_2 * t_3)), 0.5, 1.0) * 3.0)));
	else
		tmp = Float64(-Float64(fma(t_0, Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, 1.0, t_4) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0135], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$4), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.043], N[(N[(2.0 - N[(N[(N[(t$95$2 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$5 * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * 1.0 + t$95$4), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0134999999999999998

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if -0.0134999999999999998 < y < 0.042999999999999997

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 49.1

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.6

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

   9. Applied rewrites 51.6%

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

   11. Applied rewrites 51.6%

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

   12. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 48.9

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

   14. Applied rewrites 48.9%

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

   15. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 49.2

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

   17. Applied rewrites 49.2%

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

   if 0.042999999999999997 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.7%

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

   10. Taylor expanded in x around 0

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

       1. lower-sin.f64 60.6

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

   12. Applied rewrites 60.6%

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

Alternative 15: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_3 := 3 - \sqrt{5}\\ t_4 := t\_3 \cdot \cos y\\ \mathbf{if}\;y \leq -0.0135:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right) - 2}{\mathsf{fma}\left(t\_1, \cos x, t\_4\right) \cdot -1.5 - 3}\\ \mathbf{elif}\;y \leq 14:\\ \;\;\;\;\left(2 - \left(\left(t\_2 - \cos x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot \left(\left(\sin x + -0.0625 \cdot y\right) \cdot \sqrt{2}\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_2 \cdot t\_3\right), 0.5, 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(t\_0, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, 1, t\_4\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos y) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (fma (* y y) -0.5 1.0))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (* t_3 (cos y))))
   (if (<= y -0.0135)
     (/
      (- (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) t_0))) 2.0)
      (- (* (fma t_1 (cos x) t_4) -1.5) 3.0))
     (if (<= y 14.0)
       (*
        (-
         2.0
         (*
          (* (- t_2 (cos x)) (fma -0.0625 (sin x) (sin y)))
          (* (+ (sin x) (* -0.0625 y)) (sqrt 2.0))))
        (/ 1.0 (* (fma (fma t_1 (cos x) (* t_2 t_3)) 0.5 1.0) 3.0)))
       (-
        (/
         (fma
          t_0
          (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
          -2.0)
         (+ 3.0 (* 3.0 (/ (fma t_1 1.0 t_4) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(y) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma((y * y), -0.5, 1.0);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 * cos(y);
	double tmp;
	if (y <= -0.0135) {
		tmp = ((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * t_0))) - 2.0) / ((fma(t_1, cos(x), t_4) * -1.5) - 3.0);
	} else if (y <= 14.0) {
		tmp = (2.0 - (((t_2 - cos(x)) * fma(-0.0625, sin(x), sin(y))) * ((sin(x) + (-0.0625 * y)) * sqrt(2.0)))) * (1.0 / (fma(fma(t_1, cos(x), (t_2 * t_3)), 0.5, 1.0) * 3.0));
	} else {
		tmp = -(fma(t_0, (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / (3.0 + (3.0 * (fma(t_1, 1.0, t_4) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(Float64(y * y), -0.5, 1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 * cos(y))
	tmp = 0.0
	if (y <= -0.0135)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * t_0))) - 2.0) / Float64(Float64(fma(t_1, cos(x), t_4) * -1.5) - 3.0));
	elseif (y <= 14.0)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(t_2 - cos(x)) * fma(-0.0625, sin(x), sin(y))) * Float64(Float64(sin(x) + Float64(-0.0625 * y)) * sqrt(2.0)))) * Float64(1.0 / Float64(fma(fma(t_1, cos(x), Float64(t_2 * t_3)), 0.5, 1.0) * 3.0)));
	else
		tmp = Float64(-Float64(fma(t_0, Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, 1.0, t_4) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0135], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$4), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 14.0], N[(N[(2.0 - N[(N[(N[(t$95$2 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * 1.0 + t$95$4), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0134999999999999998

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if -0.0134999999999999998 < y < 14

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 49.1

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.6

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

   9. Applied rewrites 51.6%

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

   11. Applied rewrites 51.6%

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

   12. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 49.1

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

   14. Applied rewrites 49.1%

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

   if 14 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.7%

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

   10. Taylor expanded in x around 0

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

       1. lower-sin.f64 60.6

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

   12. Applied rewrites 60.6%

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

Alternative 16: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_3 := 3 - \sqrt{5}\\ t_4 := t\_3 \cdot \cos y\\ \mathbf{if}\;y \leq -0.0076:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right) - 2}{\mathsf{fma}\left(t\_1, \cos x, t\_4\right) \cdot -1.5 - 3}\\ \mathbf{elif}\;y \leq 0.0185:\\ \;\;\;\;\left(2 - \left(\left(t\_2 - \cos x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_2 \cdot t\_3\right), 0.5, 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(t\_0, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, 1, t\_4\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos y) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (fma (* y y) -0.5 1.0))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (* t_3 (cos y))))
   (if (<= y -0.0076)
     (/
      (- (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) t_0))) 2.0)
      (- (* (fma t_1 (cos x) t_4) -1.5) 3.0))
     (if (<= y 0.0185)
       (*
        (-
         2.0
         (*
          (* (- t_2 (cos x)) (fma -0.0625 (sin x) (sin y)))
          (* (sin x) (sqrt 2.0))))
        (/ 1.0 (* (fma (fma t_1 (cos x) (* t_2 t_3)) 0.5 1.0) 3.0)))
       (-
        (/
         (fma
          t_0
          (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
          -2.0)
         (+ 3.0 (* 3.0 (/ (fma t_1 1.0 t_4) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(y) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma((y * y), -0.5, 1.0);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 * cos(y);
	double tmp;
	if (y <= -0.0076) {
		tmp = ((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * t_0))) - 2.0) / ((fma(t_1, cos(x), t_4) * -1.5) - 3.0);
	} else if (y <= 0.0185) {
		tmp = (2.0 - (((t_2 - cos(x)) * fma(-0.0625, sin(x), sin(y))) * (sin(x) * sqrt(2.0)))) * (1.0 / (fma(fma(t_1, cos(x), (t_2 * t_3)), 0.5, 1.0) * 3.0));
	} else {
		tmp = -(fma(t_0, (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / (3.0 + (3.0 * (fma(t_1, 1.0, t_4) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(Float64(y * y), -0.5, 1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 * cos(y))
	tmp = 0.0
	if (y <= -0.0076)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * t_0))) - 2.0) / Float64(Float64(fma(t_1, cos(x), t_4) * -1.5) - 3.0));
	elseif (y <= 0.0185)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(t_2 - cos(x)) * fma(-0.0625, sin(x), sin(y))) * Float64(sin(x) * sqrt(2.0)))) * Float64(1.0 / Float64(fma(fma(t_1, cos(x), Float64(t_2 * t_3)), 0.5, 1.0) * 3.0)));
	else
		tmp = Float64(-Float64(fma(t_0, Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, 1.0, t_4) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0076], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$4), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0185], N[(N[(2.0 - N[(N[(N[(t$95$2 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * 1.0 + t$95$4), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00759999999999999998

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if -0.00759999999999999998 < y < 0.0184999999999999991

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 49.1

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.6

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

   9. Applied rewrites 51.6%

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

   11. Applied rewrites 51.6%

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

   12. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-sqrt.f64 49.7

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

   14. Applied rewrites 49.7%

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

   if 0.0184999999999999991 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.7%

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

   10. Taylor expanded in x around 0

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

       1. lower-sin.f64 60.6

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

   12. Applied rewrites 60.6%

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

Alternative 17: 79.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_3 := \mathsf{fma}\left(t\_1, \cos x, t\_2\right) \cdot -1.5 - 3\\ \mathbf{if}\;y \leq -32000000:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right) - 2}{t\_3}\\ \mathbf{elif}\;y \leq 750:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(t\_0, \sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, 1, t\_2\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos y) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_3 (- (* (fma t_1 (cos x) t_2) -1.5) 3.0)))
   (if (<= y -32000000.0)
     (/ (- (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) t_0))) 2.0) t_3)
     (if (<= y 750.0)
       (/
        (-
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- 1.0 (cos x)))))
         2.0)
        t_3)
       (-
        (/
         (fma
          t_0
          (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
          -2.0)
         (+ 3.0 (* 3.0 (/ (fma t_1 1.0 t_2) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(y) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (3.0 - sqrt(5.0)) * cos(y);
	double t_3 = (fma(t_1, cos(x), t_2) * -1.5) - 3.0;
	double tmp;
	if (y <= -32000000.0) {
		tmp = ((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * t_0))) - 2.0) / t_3;
	} else if (y <= 750.0) {
		tmp = ((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / t_3;
	} else {
		tmp = -(fma(t_0, (sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / (3.0 + (3.0 * (fma(t_1, 1.0, t_2) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_3 = Float64(Float64(fma(t_1, cos(x), t_2) * -1.5) - 3.0)
	tmp = 0.0
	if (y <= -32000000.0)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * t_0))) - 2.0) / t_3);
	elseif (y <= 750.0)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / t_3);
	else
		tmp = Float64(-Float64(fma(t_0, Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), -2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, 1.0, t_2) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]}, If[LessEqual[y, -32000000.0], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 750.0], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], (-N[(N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * 1.0 + t$95$2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2e7

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if -3.2e7 < y < 750

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if 750 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.7%

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

   10. Taylor expanded in x around 0

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

       1. lower-sin.f64 60.6

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

   12. Applied rewrites 60.6%

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

Alternative 18: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1 \cdot \cos y\right) \cdot -1.5 - 3\\ \mathbf{if}\;y \leq -32000000:\\ \;\;\;\;\frac{-0.0625 \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{t\_2}\\ \mathbf{elif}\;y \leq 245000000000:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(t\_1 \cdot \left(0.5 \cdot \cos y\right), 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin y) 2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (- (* (fma (- (sqrt 5.0) 1.0) (cos x) (* t_1 (cos y))) -1.5) 3.0)))
   (if (<= y -32000000.0)
     (/ (- (* -0.0625 (* t_0 (* (sqrt 2.0) (- (cos y) 1.0)))) 2.0) t_2)
     (if (<= y 245000000000.0)
       (/
        (-
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- 1.0 (cos x)))))
         2.0)
        t_2)
       (/
        (+ 2.0 (* -0.0625 (* t_0 (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma
         (* t_1 (* 0.5 (cos y)))
         3.0
         (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(y), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (fma((sqrt(5.0) - 1.0), cos(x), (t_1 * cos(y))) * -1.5) - 3.0;
	double tmp;
	if (y <= -32000000.0) {
		tmp = ((-0.0625 * (t_0 * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / t_2;
	} else if (y <= 245000000000.0) {
		tmp = ((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / t_2;
	} else {
		tmp = (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * (1.0 - cos(y)))))) / fma((t_1 * (0.5 * cos(y))), 3.0, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(y) ^ 2.0
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_1 * cos(y))) * -1.5) - 3.0)
	tmp = 0.0
	if (y <= -32000000.0)
		tmp = Float64(Float64(Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / t_2);
	elseif (y <= 245000000000.0)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / t_2);
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(Float64(t_1 * Float64(0.5 * cos(y))), 3.0, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]}, If[LessEqual[y, -32000000.0], N[(N[(N[(-0.0625 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y, 245000000000.0], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2e7

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if -3.2e7 < y < 2.45e11

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if 2.45e11 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 62.9

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

   6. Applied rewrites 62.9%

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

Alternative 19: 78.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin y}^{2}\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(t\_0, \cos x, t\_2 \cdot \cos y\right) \cdot -1.5 - 3\\ \mathbf{if}\;y \leq -32000000:\\ \;\;\;\;\frac{-0.0625 \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{t\_3}\\ \mathbf{elif}\;y \leq 245000000000:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{t\_2}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin y) 2.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (- (* (fma t_0 (cos x) (* t_2 (cos y))) -1.5) 3.0)))
   (if (<= y -32000000.0)
     (/ (- (* -0.0625 (* t_1 (* (sqrt 2.0) (- (cos y) 1.0)))) 2.0) t_3)
     (if (<= y 245000000000.0)
       (/
        (-
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- 1.0 (cos x)))))
         2.0)
        t_3)
       (/
        (+ 2.0 (* -0.0625 (* t_1 (* (sqrt 2.0) (- 1.0 (cos y))))))
        (*
         3.0
         (+ (+ 1.0 (* (/ t_0 2.0) (cos x))) (* (/ t_2 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = pow(sin(y), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = (fma(t_0, cos(x), (t_2 * cos(y))) * -1.5) - 3.0;
	double tmp;
	if (y <= -32000000.0) {
		tmp = ((-0.0625 * (t_1 * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / t_3;
	} else if (y <= 245000000000.0) {
		tmp = ((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / t_3;
	} else {
		tmp = (2.0 + (-0.0625 * (t_1 * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + ((t_2 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(Float64(fma(t_0, cos(x), Float64(t_2 * cos(y))) * -1.5) - 3.0)
	tmp = 0.0
	if (y <= -32000000.0)
		tmp = Float64(Float64(Float64(-0.0625 * Float64(t_1 * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / t_3);
	elseif (y <= 245000000000.0)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / t_3);
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_1 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(t_2 / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]}, If[LessEqual[y, -32000000.0], N[(N[(N[(-0.0625 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 245000000000.0], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2e7

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.9

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

   8. Applied rewrites 62.9%

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

   if -3.2e7 < y < 2.45e11

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if 2.45e11 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 62.9

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

   4. Applied rewrites 62.9%

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

Alternative 20: 78.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3\\ t_1 := \frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{t\_0}\\ \mathbf{if}\;y \leq -32000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 245000000000:\\ \;\;\;\;\frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (-
          (*
           (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
           -1.5)
          3.0))
        (t_1
         (/
          (-
           (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- (cos y) 1.0))))
           2.0)
          t_0)))
   (if (<= y -32000000.0)
     t_1
     (if (<= y 245000000000.0)
       (/
        (-
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- 1.0 (cos x)))))
         2.0)
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = (fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))) * -1.5) - 3.0;
	double t_1 = ((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / t_0;
	double tmp;
	if (y <= -32000000.0) {
		tmp = t_1;
	} else if (y <= 245000000000.0) {
		tmp = ((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) * -1.5) - 3.0)
	t_1 = Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / t_0)
	tmp = 0.0
	if (y <= -32000000.0)
		tmp = t_1;
	elseif (y <= 245000000000.0)
		tmp = Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -32000000.0], t$95$1, If[LessEqual[y, 245000000000.0], N[(N[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2e7 or 2.45e11 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      9. lower-cos.f64 62.9

         \[\leadsto \frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3} \]

   8. Applied rewrites 62.9%

      \[\leadsto \frac{\color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3} \]

   if -3.2e7 < y < 2.45e11

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - \color{blue}{2}}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      9. lower-cos.f64 61.3

         \[\leadsto \frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3} \]

   8. Applied rewrites 61.3%

      \[\leadsto \frac{\color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 78.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0 \cdot \cos y\right) \cdot -1.5 - 3}\\ \mathbf{if}\;x \leq -3 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_0, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (-
           (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- 1.0 (cos x)))))
           2.0)
          (- (* (fma (- (sqrt 5.0) 1.0) (cos x) (* t_0 (cos y))) -1.5) 3.0))))
   (if (<= x -3e-8)
     t_1
     (if (<= x 3.4e-6)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_0) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0))))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = ((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / ((fma((sqrt(5.0) - 1.0), cos(x), (t_0 * cos(y))) * -1.5) - 3.0);
	double tmp;
	if (x <= -3e-8) {
		tmp = t_1;
	} else if (x <= 3.4e-6) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_0), (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_0 * cos(y))) * -1.5) - 3.0))
	tmp = 0.0
	if (x <= -3e-8)
		tmp = t_1;
	elseif (x <= 3.4e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_0), Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	else
		tmp = t_1;
	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[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e-8], t$95$1, If[LessEqual[x, 3.4e-6], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999999999973e-8 or 3.40000000000000006e-6 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - \color{blue}{2}}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{-3}{2} - 3} \]

      9. lower-cos.f64 61.3

         \[\leadsto \frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3} \]

   8. Applied rewrites 61.3%

      \[\leadsto \frac{\color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3} \]

   if -2.99999999999999973e-8 < x < 3.40000000000000006e-6

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos y\right)}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right), 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   6. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 78.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0 \cdot \cos y\right), -1.5, -3\right)}\\ \mathbf{if}\;x \leq -3 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_0, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (-
           (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- 1.0 (cos x)))))
           2.0)
          (fma (fma (- (sqrt 5.0) 1.0) (cos x) (* t_0 (cos y))) -1.5 -3.0))))
   (if (<= x -3e-8)
     t_1
     (if (<= x 3.4e-6)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_0) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0))))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = ((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), (t_0 * cos(y))), -1.5, -3.0);
	double tmp;
	if (x <= -3e-8) {
		tmp = t_1;
	} else if (x <= 3.4e-6) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_0), (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_0 * cos(y))), -1.5, -3.0))
	tmp = 0.0
	if (x <= -3e-8)
		tmp = t_1;
	elseif (x <= 3.4e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_0), Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	else
		tmp = t_1;
	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[(N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.5 + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e-8], t$95$1, If[LessEqual[x, 3.4e-6], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999999999973e-8 or 3.40000000000000006e-6 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} \cdot -3 + 1 \cdot -3}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}} \cdot -3 + 1 \cdot -3} \]

      5. mult-flip N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{1}{2}\right)} \cdot -3 + 1 \cdot -3} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot -3 + 1 \cdot -3} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot -3\right)} + 1 \cdot -3} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \color{blue}{\frac{-3}{2}} + 1 \cdot -3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \color{blue}{\frac{3}{-2}} + 1 \cdot -3} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{3}{-2} + \color{blue}{-3}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{3}{-2}, -3\right)}} \]

      12. metadata-eval 99.3

          \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \color{blue}{-1.5}, -3\right)} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), -1.5, -3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{-3}{2}, -3\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - \color{blue}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{-3}{2}, -3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{-3}{2}, -3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{-3}{2}, -3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{-3}{2}, -3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{-3}{2}, -3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{-3}{2}, -3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{-3}{2}, -3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{-3}{2}, -3\right)} \]

      9. lower-cos.f64 61.3

         \[\leadsto \frac{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), -1.5, -3\right)} \]

   8. Applied rewrites 61.3%

      \[\leadsto \frac{\color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), -1.5, -3\right)} \]

   if -2.99999999999999973e-8 < x < 3.40000000000000006e-6

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos y\right)}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right), 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   6. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot \left(\frac{-1}{\mathsf{fma}\left(-0.5, t\_2, -1\right)} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_1, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_2, 0.5, 1\right)}{t\_0 \cdot 0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x))))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma (- (sqrt 5.0) 1.0) (cos x) t_1)))
   (if (<= x -3e-8)
     (* t_0 (* (/ -1.0 (fma -0.5 t_2 -1.0)) 0.3333333333333333))
     (if (<= x 1.25e-5)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_1) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0))))))
       (/ 1.0 (/ (fma t_2 0.5 1.0) (* t_0 0.3333333333333333)))))))

C

double code(double x, double y) {
	double t_0 = fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((sqrt(5.0) - 1.0), cos(x), t_1);
	double tmp;
	if (x <= -3e-8) {
		tmp = t_0 * ((-1.0 / fma(-0.5, t_2, -1.0)) * 0.3333333333333333);
	} else if (x <= 1.25e-5) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_1), (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	} else {
		tmp = 1.0 / (fma(t_2, 0.5, 1.0) / (t_0 * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(Float64(sqrt(5.0) - 1.0), cos(x), t_1)
	tmp = 0.0
	if (x <= -3e-8)
		tmp = Float64(t_0 * Float64(Float64(-1.0 / fma(-0.5, t_2, -1.0)) * 0.3333333333333333));
	elseif (x <= 1.25e-5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_1), Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	else
		tmp = Float64(1.0 / Float64(fma(t_2, 0.5, 1.0) / Float64(t_0 * 0.3333333333333333)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[x, -3e-8], N[(t$95$0 * N[(N[(-1.0 / N[(-0.5 * t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-5], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(t$95$2 * 0.5 + 1.0), $MachinePrecision] / N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999973e-8

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right) \cdot \color{blue}{\left(\frac{-1}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)} \cdot 0.3333333333333333\right)} \]

   if -2.99999999999999973e-8 < x < 1.25000000000000006e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos y\right)}, 3, 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right), 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   6. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}} \]

   if 1.25000000000000006e-5 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\color{blue}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      4. div-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}}} \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}}} \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{\color{blue}{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right) \cdot 0.3333333333333333}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(t\_1, \cos x, t\_2\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot \left(\frac{-1}{\mathsf{fma}\left(-0.5, t\_3, -1\right)} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{3 - \mathsf{fma}\left(t\_2, \cos y, t\_1\right) \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_3, 0.5, 1\right)}{t\_0 \cdot 0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x))))
          2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma t_1 (cos x) t_2)))
   (if (<= x -3e-8)
     (* t_0 (* (/ -1.0 (fma -0.5 t_3 -1.0)) 0.3333333333333333))
     (if (<= x 1.25e-5)
       (/
        (-
         2.0
         (*
          (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
          (* (- (cos y) 1.0) (sqrt 2.0))))
        (- 3.0 (* (fma t_2 (cos y) t_1) -1.5)))
       (/ 1.0 (/ (fma t_3 0.5 1.0) (* t_0 0.3333333333333333)))))))

C

double code(double x, double y) {
	double t_0 = fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(t_1, cos(x), t_2);
	double tmp;
	if (x <= -3e-8) {
		tmp = t_0 * ((-1.0 / fma(-0.5, t_3, -1.0)) * 0.3333333333333333);
	} else if (x <= 1.25e-5) {
		tmp = (2.0 - (((0.5 - (cos((y + y)) * 0.5)) * -0.0625) * ((cos(y) - 1.0) * sqrt(2.0)))) / (3.0 - (fma(t_2, cos(y), t_1) * -1.5));
	} else {
		tmp = 1.0 / (fma(t_3, 0.5, 1.0) / (t_0 * 0.3333333333333333));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(t_1, cos(x), t_2)
	tmp = 0.0
	if (x <= -3e-8)
		tmp = Float64(t_0 * Float64(Float64(-1.0 / fma(-0.5, t_3, -1.0)) * 0.3333333333333333));
	elseif (x <= 1.25e-5)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625) * Float64(Float64(cos(y) - 1.0) * sqrt(2.0)))) / Float64(3.0 - Float64(fma(t_2, cos(y), t_1) * -1.5)));
	else
		tmp = Float64(1.0 / Float64(fma(t_3, 0.5, 1.0) / Float64(t_0 * 0.3333333333333333)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[x, -3e-8], N[(t$95$0 * N[(N[(-1.0 / N[(-0.5 * t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-5], N[(N[(2.0 - N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[(N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(t$95$3 * 0.5 + 1.0), $MachinePrecision] / N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999973e-8

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right) \cdot \color{blue}{\left(\frac{-1}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)} \cdot 0.3333333333333333\right)} \]

   if -2.99999999999999973e-8 < x < 1.25000000000000006e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   8. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   10. Applied rewrites 60.1%

       \[\leadsto \frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{\color{blue}{3 - \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right) \cdot -1.5}} \]

   if 1.25000000000000006e-5 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\color{blue}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      4. div-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}}} \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}}} \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{\color{blue}{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right) \cdot 0.3333333333333333}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(t\_1, \cos x, t\_2\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot \left(\frac{-1}{\mathsf{fma}\left(-0.5, t\_3, -1\right)} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{3 - \mathsf{fma}\left(t\_2, \cos y, t\_1\right) \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{\mathsf{fma}\left(t\_3, 0.5, 1\right)}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x))))
          2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma t_1 (cos x) t_2)))
   (if (<= x -3e-8)
     (* t_0 (* (/ -1.0 (fma -0.5 t_3 -1.0)) 0.3333333333333333))
     (if (<= x 1.25e-5)
       (/
        (-
         2.0
         (*
          (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
          (* (- (cos y) 1.0) (sqrt 2.0))))
        (- 3.0 (* (fma t_2 (cos y) t_1) -1.5)))
       (* 0.3333333333333333 (/ 1.0 (/ (fma t_3 0.5 1.0) t_0)))))))

C

double code(double x, double y) {
	double t_0 = fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(t_1, cos(x), t_2);
	double tmp;
	if (x <= -3e-8) {
		tmp = t_0 * ((-1.0 / fma(-0.5, t_3, -1.0)) * 0.3333333333333333);
	} else if (x <= 1.25e-5) {
		tmp = (2.0 - (((0.5 - (cos((y + y)) * 0.5)) * -0.0625) * ((cos(y) - 1.0) * sqrt(2.0)))) / (3.0 - (fma(t_2, cos(y), t_1) * -1.5));
	} else {
		tmp = 0.3333333333333333 * (1.0 / (fma(t_3, 0.5, 1.0) / t_0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(t_1, cos(x), t_2)
	tmp = 0.0
	if (x <= -3e-8)
		tmp = Float64(t_0 * Float64(Float64(-1.0 / fma(-0.5, t_3, -1.0)) * 0.3333333333333333));
	elseif (x <= 1.25e-5)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625) * Float64(Float64(cos(y) - 1.0) * sqrt(2.0)))) / Float64(3.0 - Float64(fma(t_2, cos(y), t_1) * -1.5)));
	else
		tmp = Float64(0.3333333333333333 * Float64(1.0 / Float64(fma(t_3, 0.5, 1.0) / t_0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[x, -3e-8], N[(t$95$0 * N[(N[(-1.0 / N[(-0.5 * t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-5], N[(N[(2.0 - N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[(N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(1.0 / N[(N[(t$95$3 * 0.5 + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999973e-8

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right) \cdot \color{blue}{\left(\frac{-1}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)} \cdot 0.3333333333333333\right)} \]

   if -2.99999999999999973e-8 < x < 1.25000000000000006e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   8. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   10. Applied rewrites 60.1%

       \[\leadsto \frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{\color{blue}{3 - \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right) \cdot -1.5}} \]

   if 1.25000000000000006e-5 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1}{\color{blue}{\frac{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1}{\color{blue}{\frac{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}} \]

      4. lower-unsound-/.f64 59.0

         \[\leadsto 0.3333333333333333 \cdot \frac{1}{\frac{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}{\color{blue}{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}} \]

   6. Applied rewrites 59.0%

      \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 78.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_2 := \sqrt{5} - 1\\ t_3 := 3 - \sqrt{5}\\ t_4 := \mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_2, \cos x, t\_3\right), -1\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.0625 \cdot t\_1, t\_0, 2\right) \cdot \left(\frac{-1}{t\_4} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{3 - \mathsf{fma}\left(t\_3, \cos y, t\_2\right) \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625 \cdot t\_1, t\_0, -2\right) \cdot 0.3333333333333333}{t\_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (fma -0.5 (fma t_2 (cos x) t_3) -1.0)))
   (if (<= x -3e-8)
     (* (fma (* -0.0625 t_1) t_0 2.0) (* (/ -1.0 t_4) 0.3333333333333333))
     (if (<= x 1.25e-5)
       (/
        (-
         2.0
         (*
          (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
          (* (- (cos y) 1.0) (sqrt 2.0))))
        (- 3.0 (* (fma t_3 (cos y) t_2) -1.5)))
       (/ (* (fma (* 0.0625 t_1) t_0 -2.0) 0.3333333333333333) t_4)))))

C

double code(double x, double y) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * x)));
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = fma(-0.5, fma(t_2, cos(x), t_3), -1.0);
	double tmp;
	if (x <= -3e-8) {
		tmp = fma((-0.0625 * t_1), t_0, 2.0) * ((-1.0 / t_4) * 0.3333333333333333);
	} else if (x <= 1.25e-5) {
		tmp = (2.0 - (((0.5 - (cos((y + y)) * 0.5)) * -0.0625) * ((cos(y) - 1.0) * sqrt(2.0)))) / (3.0 - (fma(t_3, cos(y), t_2) * -1.5));
	} else {
		tmp = (fma((0.0625 * t_1), t_0, -2.0) * 0.3333333333333333) / t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = fma(-0.5, fma(t_2, cos(x), t_3), -1.0)
	tmp = 0.0
	if (x <= -3e-8)
		tmp = Float64(fma(Float64(-0.0625 * t_1), t_0, 2.0) * Float64(Float64(-1.0 / t_4) * 0.3333333333333333));
	elseif (x <= 1.25e-5)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625) * Float64(Float64(cos(y) - 1.0) * sqrt(2.0)))) / Float64(3.0 - Float64(fma(t_3, cos(y), t_2) * -1.5)));
	else
		tmp = Float64(Float64(fma(Float64(0.0625 * t_1), t_0, -2.0) * 0.3333333333333333) / t_4);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -3e-8], N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] * N[(N[(-1.0 / t$95$4), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-5], N[(N[(2.0 - N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[(N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * t$95$1), $MachinePrecision] * t$95$0 + -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / t$95$4), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999973e-8

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right) \cdot \color{blue}{\left(\frac{-1}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)} \cdot 0.3333333333333333\right)} \]

   if -2.99999999999999973e-8 < x < 1.25000000000000006e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   8. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   10. Applied rewrites 60.1%

       \[\leadsto \frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{\color{blue}{3 - \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right) \cdot -1.5}} \]

   if 1.25000000000000006e-5 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right)\\ t_3 := \mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), -1\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-8}:\\ \;\;\;\;0.3333333333333333 \cdot \left(t\_2 \cdot \frac{1}{t\_3}\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{3 - \mathsf{fma}\left(t\_1, \cos y, t\_0\right) \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 \cdot 0.3333333333333333}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (fma
          (* 0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x))))
          -2.0))
        (t_3 (fma -0.5 (fma t_0 (cos x) t_1) -1.0)))
   (if (<= x -3e-8)
     (* 0.3333333333333333 (* t_2 (/ 1.0 t_3)))
     (if (<= x 1.25e-5)
       (/
        (-
         2.0
         (*
          (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
          (* (- (cos y) 1.0) (sqrt 2.0))))
        (- 3.0 (* (fma t_1 (cos y) t_0) -1.5)))
       (/ (* t_2 0.3333333333333333) t_3)))))

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((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), -2.0);
	double t_3 = fma(-0.5, fma(t_0, cos(x), t_1), -1.0);
	double tmp;
	if (x <= -3e-8) {
		tmp = 0.3333333333333333 * (t_2 * (1.0 / t_3));
	} else if (x <= 1.25e-5) {
		tmp = (2.0 - (((0.5 - (cos((y + y)) * 0.5)) * -0.0625) * ((cos(y) - 1.0) * sqrt(2.0)))) / (3.0 - (fma(t_1, cos(y), t_0) * -1.5));
	} else {
		tmp = (t_2 * 0.3333333333333333) / t_3;
	}
	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(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), -2.0)
	t_3 = fma(-0.5, fma(t_0, cos(x), t_1), -1.0)
	tmp = 0.0
	if (x <= -3e-8)
		tmp = Float64(0.3333333333333333 * Float64(t_2 * Float64(1.0 / t_3)));
	elseif (x <= 1.25e-5)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625) * Float64(Float64(cos(y) - 1.0) * sqrt(2.0)))) / Float64(3.0 - Float64(fma(t_1, cos(y), t_0) * -1.5)));
	else
		tmp = Float64(Float64(t_2 * 0.3333333333333333) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -3e-8], N[(0.3333333333333333 * N[(t$95$2 * N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-5], N[(N[(2.0 - N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * 0.3333333333333333), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999973e-8

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.0%

      \[\leadsto 0.3333333333333333 \cdot \left(\mathsf{fma}\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}}\right) \]

   if -2.99999999999999973e-8 < x < 1.25000000000000006e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   8. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   10. Applied rewrites 60.1%

       \[\leadsto \frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{\color{blue}{3 - \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right) \cdot -1.5}} \]

   if 1.25000000000000006e-5 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_3 := 3 - \sqrt{5}\\ t_4 := \mathsf{fma}\left(t\_1, \cos x, t\_3\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_2, t\_0, 2\right)}{\mathsf{fma}\left(t\_4, 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{3 - \mathsf{fma}\left(t\_3, \cos y, t\_1\right) \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625 \cdot t\_2, t\_0, -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, t\_4, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (fma t_1 (cos x) t_3)))
   (if (<= x -3e-8)
     (* (/ (fma (* -0.0625 t_2) t_0 2.0) (fma t_4 0.5 1.0)) 0.3333333333333333)
     (if (<= x 1.25e-5)
       (/
        (-
         2.0
         (*
          (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
          (* (- (cos y) 1.0) (sqrt 2.0))))
        (- 3.0 (* (fma t_3 (cos y) t_1) -1.5)))
       (/
        (* (fma (* 0.0625 t_2) t_0 -2.0) 0.3333333333333333)
        (fma -0.5 t_4 -1.0))))))

C

double code(double x, double y) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * x)));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (cos(x) - 1.0) * sqrt(2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = fma(t_1, cos(x), t_3);
	double tmp;
	if (x <= -3e-8) {
		tmp = (fma((-0.0625 * t_2), t_0, 2.0) / fma(t_4, 0.5, 1.0)) * 0.3333333333333333;
	} else if (x <= 1.25e-5) {
		tmp = (2.0 - (((0.5 - (cos((y + y)) * 0.5)) * -0.0625) * ((cos(y) - 1.0) * sqrt(2.0)))) / (3.0 - (fma(t_3, cos(y), t_1) * -1.5));
	} else {
		tmp = (fma((0.0625 * t_2), t_0, -2.0) * 0.3333333333333333) / fma(-0.5, t_4, -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = fma(t_1, cos(x), t_3)
	tmp = 0.0
	if (x <= -3e-8)
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_2), t_0, 2.0) / fma(t_4, 0.5, 1.0)) * 0.3333333333333333);
	elseif (x <= 1.25e-5)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625) * Float64(Float64(cos(y) - 1.0) * sqrt(2.0)))) / Float64(3.0 - Float64(fma(t_3, cos(y), t_1) * -1.5)));
	else
		tmp = Float64(Float64(fma(Float64(0.0625 * t_2), t_0, -2.0) * 0.3333333333333333) / fma(-0.5, t_4, -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[x, -3e-8], N[(N[(N[(N[(-0.0625 * t$95$2), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(t$95$4 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.25e-5], N[(N[(2.0 - N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[(N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * t$95$2), $MachinePrecision] * t$95$0 + -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999973e-8

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]

   if -2.99999999999999973e-8 < x < 1.25000000000000006e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   8. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   10. Applied rewrites 60.1%

       \[\leadsto \frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{\color{blue}{3 - \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right) \cdot -1.5}} \]

   if 1.25000000000000006e-5 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\mathsf{fma}\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), -1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -3 \cdot 10^{-8}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{3 - \mathsf{fma}\left(t\_1, \cos y, t\_0\right) \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (*
          (/
           (fma
            (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
            (- 0.5 (* 0.5 (cos (* 2.0 x))))
            2.0)
           (fma (fma t_0 (cos x) t_1) 0.5 1.0))
          0.3333333333333333)))
   (if (<= x -3e-8)
     t_2
     (if (<= x 1.25e-5)
       (/
        (-
         2.0
         (*
          (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
          (* (- (cos y) 1.0) (sqrt 2.0))))
        (- 3.0 (* (fma t_1 (cos y) t_0) -1.5)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -3e-8) {
		tmp = t_2;
	} else if (x <= 1.25e-5) {
		tmp = (2.0 - (((0.5 - (cos((y + y)) * 0.5)) * -0.0625) * ((cos(y) - 1.0) * sqrt(2.0)))) / (3.0 - (fma(t_1, cos(y), t_0) * -1.5));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(fma(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -3e-8)
		tmp = t_2;
	elseif (x <= 1.25e-5)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625) * Float64(Float64(cos(y) - 1.0) * sqrt(2.0)))) / Float64(3.0 - Float64(fma(t_1, cos(y), t_0) * -1.5)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -3e-8], t$95$2, If[LessEqual[x, 1.25e-5], N[(N[(2.0 - N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999999999973e-8 or 1.25000000000000006e-5 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \cdot 0.3333333333333333} \]

   if -2.99999999999999973e-8 < x < 1.25000000000000006e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   8. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

   10. Applied rewrites 60.1%

       \[\leadsto \frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{\color{blue}{3 - \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right) \cdot -1.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 60.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{3 - \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right) \cdot -1.5} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (-
   2.0
   (*
    (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
    (* (- (cos y) 1.0) (sqrt 2.0))))
  (- 3.0 (* (fma (- 3.0 (sqrt 5.0)) (cos y) (- (sqrt 5.0) 1.0)) -1.5))))

C

double code(double x, double y) {
	return (2.0 - (((0.5 - (cos((y + y)) * 0.5)) * -0.0625) * ((cos(y) - 1.0) * sqrt(2.0)))) / (3.0 - (fma((3.0 - sqrt(5.0)), cos(y), (sqrt(5.0) - 1.0)) * -1.5));
}

Julia

function code(x, y)
	return Float64(Float64(2.0 - Float64(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625) * Float64(Float64(cos(y) - 1.0) * sqrt(2.0)))) / Float64(3.0 - Float64(fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(sqrt(5.0) - 1.0)) * -1.5)))
end

Wolfram

code[x_, y_] := N[(N[(2.0 - N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

5. Applied rewrites 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

8. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

10. Applied rewrites 60.1%

    \[\leadsto \frac{2 - \left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\left(\cos y - 1\right) \cdot \sqrt{2}\right)}{\color{blue}{3 - \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right) \cdot -1.5}} \]
11. Add Preprocessing

Alternative 31: 60.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(\cos y - 1\right) \cdot \sqrt{2}, -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), -1.5, -3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
   (* (- (cos y) 1.0) (sqrt 2.0))
   -2.0)
  (fma (fma (- 3.0 (sqrt 5.0)) (cos y) (- (sqrt 5.0) 1.0)) -1.5 -3.0)))

C

double code(double x, double y) {
	return fma(((0.5 - (cos((y + y)) * 0.5)) * -0.0625), ((cos(y) - 1.0) * sqrt(2.0)), -2.0) / fma(fma((3.0 - sqrt(5.0)), cos(y), (sqrt(5.0) - 1.0)), -1.5, -3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), Float64(Float64(cos(y) - 1.0) * sqrt(2.0)), -2.0) / fma(fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(sqrt(5.0) - 1.0)), -1.5, -3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[y], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * -1.5 + -3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

5. Applied rewrites 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

8. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

10. Applied rewrites 60.1%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(\cos y - 1\right) \cdot \sqrt{2}, -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), -1.5, -3\right)}} \]
11. Add Preprocessing

Alternative 32: 42.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   2.0
   (+
    1.0
    (fma 0.5 (* (cos x) (- (sqrt 5.0) 1.0)) (* 0.5 (- 3.0 (sqrt 5.0))))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * (2.0 / (1.0 + fma(0.5, (cos(x) * (sqrt(5.0) - 1.0)), (0.5 * (3.0 - sqrt(5.0))))));
}

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(2.0 / Float64(1.0 + fma(0.5, Float64(cos(x) * Float64(sqrt(5.0) - 1.0)), Float64(0.5 * Float64(3.0 - sqrt(5.0)))))))
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(2.0 / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 59.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \cdot \frac{2}{\color{blue}{1} + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 42.9%

   \[\leadsto 0.3333333333333333 \cdot \frac{2}{\color{blue}{1} + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
8. Add Preprocessing

Alternative 33: 42.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  -2.0
  (- (* -1.5 (- (+ (sqrt 5.0) (* (cos y) (- 3.0 (sqrt 5.0)))) 1.0)) 3.0)))

C

double code(double x, double y) {
	return -2.0 / ((-1.5 * ((sqrt(5.0) + (cos(y) * (3.0 - sqrt(5.0)))) - 1.0)) - 3.0);
}

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) / (((-1.5d0) * ((sqrt(5.0d0) + (cos(y) * (3.0d0 - sqrt(5.0d0)))) - 1.0d0)) - 3.0d0)
end function

Java

public static double code(double x, double y) {
	return -2.0 / ((-1.5 * ((Math.sqrt(5.0) + (Math.cos(y) * (3.0 - Math.sqrt(5.0)))) - 1.0)) - 3.0);
}

Python

def code(x, y):
	return -2.0 / ((-1.5 * ((math.sqrt(5.0) + (math.cos(y) * (3.0 - math.sqrt(5.0)))) - 1.0)) - 3.0)

Julia

function code(x, y)
	return Float64(-2.0 / Float64(Float64(-1.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) - 1.0)) - 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = -2.0 / ((-1.5 * ((sqrt(5.0) + (cos(y) * (3.0 - sqrt(5.0)))) - 1.0)) - 3.0);
end

Wolfram

code[x_, y_] := N[(-2.0 / N[(N[(-1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

5. Applied rewrites 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

8. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{-2}{\color{blue}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)} - 3} \]
10. Step-by-step derivation

11. Applied rewrites 42.5%

    \[\leadsto \frac{-2}{\color{blue}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)} - 3} \]
12. Add Preprocessing

Alternative 34: 40.5% accurate, 316.7× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{-3 \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1\right)}} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{-3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + -3 \cdot 1} \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot 1}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3} \cdot 1} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} - \color{blue}{3}} \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\left(\mathsf{neg}\left(3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\right)\right)} - 3} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}\right)\right) - 3} \]

5. Applied rewrites 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\cos y - \cos x, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), -2\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot -1.5 - 3}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{-3}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

8. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\frac{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{-1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) - 3}} \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{1}{3} \]
10. Step-by-step derivation

11. Applied rewrites 40.5%

    \[\leadsto 0.3333333333333333 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(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))))))