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

Percentage Accurate: 99.3% → 99.3%

Time: 9.5s

Alternatives: 31

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}\\ \mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot t\_0}, \frac{0.6666666666666666}{t\_0 + 1}\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          2.0)))
   (fma
    (- (cos x) (cos y))
    (/
     (*
      (fma (sin x) -0.0625 (sin y))
      (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
     (+ 3.0 (* 3.0 t_0)))
    (/ 0.6666666666666666 (+ t_0 1.0)))))

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))) / 2.0;
	return fma((cos(x) - cos(y)), ((fma(sin(x), -0.0625, sin(y)) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))) / (3.0 + (3.0 * t_0))), (0.6666666666666666 / (t_0 + 1.0)));
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) / 2.0)
	return fma(Float64(cos(x) - cos(y)), Float64(Float64(fma(sin(x), -0.0625, sin(y)) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))) / Float64(3.0 + Float64(3.0 * t_0))), Float64(0.6666666666666666 / Float64(t_0 + 1.0)))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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] / 2.0), $MachinePrecision]}, N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(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] / N[(3.0 + N[(3.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]
4. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

5. Applied rewrites 99.2%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

5. Applied rewrites 99.2%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

5. Applied rewrites 99.3%

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

Alternative 5: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_1, \sin x \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(t\_0, 1.5, 3\right)}\\ \mathbf{if}\;x \leq -0.056:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1 \cdot \sqrt{2}, \left(\sin y + x \cdot \left(0.010416666666666666 \cdot {x}^{2} - 0.0625\right)\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\frac{t\_0}{2} + 1}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 x) (cos y)))
        (t_2
         (/
          (fma
           (* (fma -0.0625 (sin x) (sin y)) t_1)
           (* (sin x) (sqrt 2.0))
           2.0)
          (fma t_0 1.5 3.0))))
   (if (<= x -0.056)
     t_2
     (if (<= x 0.23)
       (/
        (/
         (fma
          (* t_1 (sqrt 2.0))
          (*
           (+ (sin y) (* x (- (* 0.010416666666666666 (pow x 2.0)) 0.0625)))
           (fma (sin y) -0.0625 (sin x)))
          2.0)
         (+ (/ t_0 2.0) 1.0))
        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(x) - cos(y);
	double t_2 = fma((fma(-0.0625, sin(x), sin(y)) * t_1), (sin(x) * sqrt(2.0)), 2.0) / fma(t_0, 1.5, 3.0);
	double tmp;
	if (x <= -0.056) {
		tmp = t_2;
	} else if (x <= 0.23) {
		tmp = (fma((t_1 * sqrt(2.0)), ((sin(y) + (x * ((0.010416666666666666 * pow(x, 2.0)) - 0.0625))) * fma(sin(y), -0.0625, sin(x))), 2.0) / ((t_0 / 2.0) + 1.0)) / 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(x) - cos(y))
	t_2 = Float64(fma(Float64(fma(-0.0625, sin(x), sin(y)) * t_1), Float64(sin(x) * sqrt(2.0)), 2.0) / fma(t_0, 1.5, 3.0))
	tmp = 0.0
	if (x <= -0.056)
		tmp = t_2;
	elseif (x <= 0.23)
		tmp = Float64(Float64(fma(Float64(t_1 * sqrt(2.0)), Float64(Float64(sin(y) + Float64(x * Float64(Float64(0.010416666666666666 * (x ^ 2.0)) - 0.0625))) * fma(sin(y), -0.0625, sin(x))), 2.0) / Float64(Float64(t_0 / 2.0) + 1.0)) / 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[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.056], t$95$2, If[LessEqual[x, 0.23], N[(N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(x * N[(N[(0.010416666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0560000000000000012 or 0.23000000000000001 < 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

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

      1. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   6. Applied rewrites 50.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.7

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

   9. Applied rewrites 63.7%

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

   if -0.0560000000000000012 < x < 0.23000000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 50.5

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

   6. Applied rewrites 50.5%

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

Alternative 6: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1.5, 3\right)\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_1, \sin x \cdot \sqrt{2}, 2\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.056:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;\frac{1}{\frac{t\_0}{\mathsf{fma}\left(\left(t\_1 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sin y + x \cdot \left(0.010416666666666666 \cdot {x}^{2} - 0.0625\right), 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          1.5
          3.0))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (/
          (fma
           (* (fma -0.0625 (sin x) (sin y)) t_1)
           (* (sin x) (sqrt 2.0))
           2.0)
          t_0)))
   (if (<= x -0.056)
     t_2
     (if (<= x 0.23)
       (/
        1.0
        (/
         t_0
         (fma
          (* (* t_1 (sqrt 2.0)) (fma (sin y) -0.0625 (sin x)))
          (+ (sin y) (* x (- (* 0.010416666666666666 (pow x 2.0)) 0.0625)))
          2.0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.5, 3.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = fma((fma(-0.0625, sin(x), sin(y)) * t_1), (sin(x) * sqrt(2.0)), 2.0) / t_0;
	double tmp;
	if (x <= -0.056) {
		tmp = t_2;
	} else if (x <= 0.23) {
		tmp = 1.0 / (t_0 / fma(((t_1 * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), (sin(y) + (x * ((0.010416666666666666 * pow(x, 2.0)) - 0.0625))), 2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(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(x) - cos(y))
	t_2 = Float64(fma(Float64(fma(-0.0625, sin(x), sin(y)) * t_1), Float64(sin(x) * sqrt(2.0)), 2.0) / t_0)
	tmp = 0.0
	if (x <= -0.056)
		tmp = t_2;
	elseif (x <= 0.23)
		tmp = Float64(1.0 / Float64(t_0 / fma(Float64(Float64(t_1 * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), Float64(sin(y) + Float64(x * Float64(Float64(0.010416666666666666 * (x ^ 2.0)) - 0.0625))), 2.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.056], t$95$2, If[LessEqual[x, 0.23], N[(1.0 / N[(t$95$0 / N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(x * N[(N[(0.010416666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0560000000000000012 or 0.23000000000000001 < 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

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

      1. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   6. Applied rewrites 50.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.7

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

   9. Applied rewrites 63.7%

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

   if -0.0560000000000000012 < x < 0.23000000000000001

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 50.5

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

   8. Applied rewrites 50.5%

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

Alternative 7: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)\\ t_1 := x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\\ t_2 := \cos x - \cos y\\ t_3 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_2, \sin x \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(t\_0, 1.5, 3\right)}\\ \mathbf{if}\;x \leq -0.056:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, \mathsf{fma}\left(t\_1, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, t\_1\right), 2\right)}{\frac{t\_0}{2} + 1}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))))
        (t_1 (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0)))))
        (t_2 (- (cos x) (cos y)))
        (t_3
         (/
          (fma
           (* (fma -0.0625 (sin x) (sin y)) t_2)
           (* (sin x) (sqrt 2.0))
           2.0)
          (fma t_0 1.5 3.0))))
   (if (<= x -0.056)
     t_3
     (if (<= x 0.22)
       (/
        (/
         (fma
          (* t_2 (sqrt 2.0))
          (* (fma t_1 -0.0625 (sin y)) (fma (sin y) -0.0625 t_1))
          2.0)
         (+ (/ t_0 2.0) 1.0))
        3.0)
       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)));
	double t_1 = x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)));
	double t_2 = cos(x) - cos(y);
	double t_3 = fma((fma(-0.0625, sin(x), sin(y)) * t_2), (sin(x) * sqrt(2.0)), 2.0) / fma(t_0, 1.5, 3.0);
	double tmp;
	if (x <= -0.056) {
		tmp = t_3;
	} else if (x <= 0.22) {
		tmp = (fma((t_2 * sqrt(2.0)), (fma(t_1, -0.0625, sin(y)) * fma(sin(y), -0.0625, t_1)), 2.0) / ((t_0 / 2.0) + 1.0)) / 3.0;
	} else {
		tmp = t_3;
	}
	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(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(fma(Float64(fma(-0.0625, sin(x), sin(y)) * t_2), Float64(sin(x) * sqrt(2.0)), 2.0) / fma(t_0, 1.5, 3.0))
	tmp = 0.0
	if (x <= -0.056)
		tmp = t_3;
	elseif (x <= 0.22)
		tmp = Float64(Float64(fma(Float64(t_2 * sqrt(2.0)), Float64(fma(t_1, -0.0625, sin(y)) * fma(sin(y), -0.0625, t_1)), 2.0) / Float64(Float64(t_0 / 2.0) + 1.0)) / 3.0);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * 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[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.056], t$95$3, If[LessEqual[x, 0.22], N[(N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + t$95$1), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0560000000000000012 or 0.220000000000000001 < 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

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

      1. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   6. Applied rewrites 50.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.7

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

   9. Applied rewrites 63.7%

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

   if -0.0560000000000000012 < x < 0.220000000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.5

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.7

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

   9. Applied rewrites 50.7%

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

Alternative 8: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1.5, 3\right)\\ t_1 := x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\\ t_2 := \cos x - \cos y\\ t_3 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_2, \sin x \cdot \sqrt{2}, 2\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.056:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{1}{\frac{t\_0}{\mathsf{fma}\left(\left(t\_2 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, t\_1\right), \mathsf{fma}\left(-0.0625, t\_1, \sin y\right), 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          1.5
          3.0))
        (t_1 (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0)))))
        (t_2 (- (cos x) (cos y)))
        (t_3
         (/
          (fma
           (* (fma -0.0625 (sin x) (sin y)) t_2)
           (* (sin x) (sqrt 2.0))
           2.0)
          t_0)))
   (if (<= x -0.056)
     t_3
     (if (<= x 0.22)
       (/
        1.0
        (/
         t_0
         (fma
          (* (* t_2 (sqrt 2.0)) (fma (sin y) -0.0625 t_1))
          (fma -0.0625 t_1 (sin y))
          2.0)))
       t_3))))

C

double code(double x, double y) {
	double t_0 = fma(fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.5, 3.0);
	double t_1 = x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)));
	double t_2 = cos(x) - cos(y);
	double t_3 = fma((fma(-0.0625, sin(x), sin(y)) * t_2), (sin(x) * sqrt(2.0)), 2.0) / t_0;
	double tmp;
	if (x <= -0.056) {
		tmp = t_3;
	} else if (x <= 0.22) {
		tmp = 1.0 / (t_0 / fma(((t_2 * sqrt(2.0)) * fma(sin(y), -0.0625, t_1)), fma(-0.0625, t_1, sin(y)), 2.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(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(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(fma(Float64(fma(-0.0625, sin(x), sin(y)) * t_2), Float64(sin(x) * sqrt(2.0)), 2.0) / t_0)
	tmp = 0.0
	if (x <= -0.056)
		tmp = t_3;
	elseif (x <= 0.22)
		tmp = Float64(1.0 / Float64(t_0 / fma(Float64(Float64(t_2 * sqrt(2.0)) * fma(sin(y), -0.0625, t_1)), fma(-0.0625, t_1, sin(y)), 2.0)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.056], t$95$3, If[LessEqual[x, 0.22], N[(1.0 / N[(t$95$0 / N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$1 + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0560000000000000012 or 0.220000000000000001 < 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

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

      1. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   6. Applied rewrites 50.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.7

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

   9. Applied rewrites 63.7%

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

   if -0.0560000000000000012 < x < 0.220000000000000001

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.4

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.7

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

   11. Applied rewrites 50.7%

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

Alternative 9: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_1, \sin x \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(t\_0, 1.5, 3\right)}\\ \mathbf{if}\;x \leq -0.054:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1 \cdot \sqrt{2}, \left(\sin y + -0.0625 \cdot x\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\frac{t\_0}{2} + 1}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 x) (cos y)))
        (t_2
         (/
          (fma
           (* (fma -0.0625 (sin x) (sin y)) t_1)
           (* (sin x) (sqrt 2.0))
           2.0)
          (fma t_0 1.5 3.0))))
   (if (<= x -0.054)
     t_2
     (if (<= x 0.22)
       (/
        (/
         (fma
          (* t_1 (sqrt 2.0))
          (* (+ (sin y) (* -0.0625 x)) (fma (sin y) -0.0625 (sin x)))
          2.0)
         (+ (/ t_0 2.0) 1.0))
        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(x) - cos(y);
	double t_2 = fma((fma(-0.0625, sin(x), sin(y)) * t_1), (sin(x) * sqrt(2.0)), 2.0) / fma(t_0, 1.5, 3.0);
	double tmp;
	if (x <= -0.054) {
		tmp = t_2;
	} else if (x <= 0.22) {
		tmp = (fma((t_1 * sqrt(2.0)), ((sin(y) + (-0.0625 * x)) * fma(sin(y), -0.0625, sin(x))), 2.0) / ((t_0 / 2.0) + 1.0)) / 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(x) - cos(y))
	t_2 = Float64(fma(Float64(fma(-0.0625, sin(x), sin(y)) * t_1), Float64(sin(x) * sqrt(2.0)), 2.0) / fma(t_0, 1.5, 3.0))
	tmp = 0.0
	if (x <= -0.054)
		tmp = t_2;
	elseif (x <= 0.22)
		tmp = Float64(Float64(fma(Float64(t_1 * sqrt(2.0)), Float64(Float64(sin(y) + Float64(-0.0625 * x)) * fma(sin(y), -0.0625, sin(x))), 2.0) / Float64(Float64(t_0 / 2.0) + 1.0)) / 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[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.054], t$95$2, If[LessEqual[x, 0.22], N[(N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(-0.0625 * x), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0539999999999999994 or 0.220000000000000001 < 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

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

      1. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   6. Applied rewrites 50.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.7

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

   9. Applied rewrites 63.7%

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

   if -0.0539999999999999994 < x < 0.220000000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 50.9

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

   6. Applied rewrites 50.9%

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

Alternative 10: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos x - \cos y\\ t_3 := \sqrt{5} - 1\\ t_4 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot t\_2, \sin x \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_3, \cos x, t\_1 \cdot \cos y\right), 1.5, 3\right)}\\ \mathbf{if}\;x \leq -0.056:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_0, -0.0625, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, t\_0\right) \cdot \left(t\_2 \cdot \sqrt{2}\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_3, \cos y \cdot t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4
         (/
          (fma
           (* (fma -0.0625 (sin x) (sin y)) t_2)
           (* (sin x) (sqrt 2.0))
           2.0)
          (fma (fma t_3 (cos x) (* t_1 (cos y))) 1.5 3.0))))
   (if (<= x -0.056)
     t_4
     (if (<= x 0.22)
       (*
        (fma
         (fma t_0 -0.0625 (sin y))
         (* (fma (sin y) -0.0625 t_0) (* t_2 (sqrt 2.0)))
         2.0)
        (/ 1.0 (fma 1.5 (fma (cos x) t_3 (* (cos y) t_1)) 3.0)))
       t_4))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = fma((fma(-0.0625, sin(x), sin(y)) * t_2), (sin(x) * sqrt(2.0)), 2.0) / fma(fma(t_3, cos(x), (t_1 * cos(y))), 1.5, 3.0);
	double tmp;
	if (x <= -0.056) {
		tmp = t_4;
	} else if (x <= 0.22) {
		tmp = fma(fma(t_0, -0.0625, sin(y)), (fma(sin(y), -0.0625, t_0) * (t_2 * sqrt(2.0))), 2.0) * (1.0 / fma(1.5, fma(cos(x), t_3, (cos(y) * t_1)), 3.0));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = Float64(fma(Float64(fma(-0.0625, sin(x), sin(y)) * t_2), Float64(sin(x) * sqrt(2.0)), 2.0) / fma(fma(t_3, cos(x), Float64(t_1 * cos(y))), 1.5, 3.0))
	tmp = 0.0
	if (x <= -0.056)
		tmp = t_4;
	elseif (x <= 0.22)
		tmp = Float64(fma(fma(t_0, -0.0625, sin(y)), Float64(fma(sin(y), -0.0625, t_0) * Float64(t_2 * sqrt(2.0))), 2.0) * Float64(1.0 / fma(1.5, fma(cos(x), t_3, Float64(cos(y) * t_1)), 3.0)));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$3 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.056], t$95$4, If[LessEqual[x, 0.22], N[(N[(N[(t$95$0 * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + t$95$0), $MachinePrecision] * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$3 + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0560000000000000012 or 0.220000000000000001 < 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

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

      1. lower-*.f64 50.7

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

   4. Applied rewrites 50.7%

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

   6. Applied rewrites 50.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.7

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

   9. Applied rewrites 63.7%

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

   if -0.0560000000000000012 < x < 0.220000000000000001

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.4

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.7

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

   11. Applied rewrites 50.7%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. div-flip-rev N/A

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

   13. Applied rewrites 50.7%

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

Alternative 11: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_2 := \sqrt{5} - 1\\ t_3 := \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right)\\ t_4 := \left(\cos x - \cos y\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -0.056:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_3, 1.5, 3\right)}{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}}\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_1, -0.0625, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, t\_1\right) \cdot t\_4, 2\right) \cdot \frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_4, -0.0625 \cdot {\sin x}^{2}, 2\right)}{\frac{t\_3}{2} + 1}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (fma t_2 (cos x) (* t_0 (cos y))))
        (t_4 (* (- (cos x) (cos y)) (sqrt 2.0))))
   (if (<= x -0.056)
     (/
      1.0
      (/
       (fma t_3 1.5 3.0)
       (fma
        (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0)))
        (fma -0.0625 (sin x) (sin y))
        2.0)))
     (if (<= x 0.23)
       (*
        (fma (fma t_1 -0.0625 (sin y)) (* (fma (sin y) -0.0625 t_1) t_4) 2.0)
        (/ 1.0 (fma 1.5 (fma (cos x) t_2 (* (cos y) t_0)) 3.0)))
       (/
        (/ (fma t_4 (* -0.0625 (pow (sin x) 2.0)) 2.0) (+ (/ t_3 2.0) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma(t_2, cos(x), (t_0 * cos(y)));
	double t_4 = (cos(x) - cos(y)) * sqrt(2.0);
	double tmp;
	if (x <= -0.056) {
		tmp = 1.0 / (fma(t_3, 1.5, 3.0) / fma((sin(x) * (sqrt(2.0) * (cos(x) - 1.0))), fma(-0.0625, sin(x), sin(y)), 2.0));
	} else if (x <= 0.23) {
		tmp = fma(fma(t_1, -0.0625, sin(y)), (fma(sin(y), -0.0625, t_1) * t_4), 2.0) * (1.0 / fma(1.5, fma(cos(x), t_2, (cos(y) * t_0)), 3.0));
	} else {
		tmp = (fma(t_4, (-0.0625 * pow(sin(x), 2.0)), 2.0) / ((t_3 / 2.0) + 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = fma(t_2, cos(x), Float64(t_0 * cos(y)))
	t_4 = Float64(Float64(cos(x) - cos(y)) * sqrt(2.0))
	tmp = 0.0
	if (x <= -0.056)
		tmp = Float64(1.0 / Float64(fma(t_3, 1.5, 3.0) / fma(Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))), fma(-0.0625, sin(x), sin(y)), 2.0)));
	elseif (x <= 0.23)
		tmp = Float64(fma(fma(t_1, -0.0625, sin(y)), Float64(fma(sin(y), -0.0625, t_1) * t_4), 2.0) * Float64(1.0 / fma(1.5, fma(cos(x), t_2, Float64(cos(y) * t_0)), 3.0)));
	else
		tmp = Float64(Float64(fma(t_4, Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(Float64(t_3 / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.056], N[(1.0 / N[(N[(t$95$3 * 1.5 + 3.0), $MachinePrecision] / N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.23], N[(N[(N[(t$95$1 * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + t$95$1), $MachinePrecision] * t$95$4), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$4 * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$3 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0560000000000000012

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -0.0560000000000000012 < x < 0.23000000000000001

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.4

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.7

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

   11. Applied rewrites 50.7%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. div-flip-rev N/A

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

   13. Applied rewrites 50.7%

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

   if 0.23000000000000001 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.0

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

   6. Applied rewrites 62.0%

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

Alternative 12: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_2 := \sqrt{5} - 1\\ t_3 := \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right)\\ t_4 := \left(\cos x - \cos y\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -0.056:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_3, 1.5, 3\right)}{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}}\\ \mathbf{elif}\;x \leq 0.23:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, -0.0625, \sin y\right), \mathsf{fma}\left(\sin y, -0.0625, t\_1\right) \cdot t\_4, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_4, -0.0625 \cdot {\sin x}^{2}, 2\right)}{\frac{t\_3}{2} + 1}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (fma t_2 (cos x) (* t_0 (cos y))))
        (t_4 (* (- (cos x) (cos y)) (sqrt 2.0))))
   (if (<= x -0.056)
     (/
      1.0
      (/
       (fma t_3 1.5 3.0)
       (fma
        (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0)))
        (fma -0.0625 (sin x) (sin y))
        2.0)))
     (if (<= x 0.23)
       (/
        (fma (fma t_1 -0.0625 (sin y)) (* (fma (sin y) -0.0625 t_1) t_4) 2.0)
        (fma 1.5 (fma (cos x) t_2 (* (cos y) t_0)) 3.0))
       (/
        (/ (fma t_4 (* -0.0625 (pow (sin x) 2.0)) 2.0) (+ (/ t_3 2.0) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma(t_2, cos(x), (t_0 * cos(y)));
	double t_4 = (cos(x) - cos(y)) * sqrt(2.0);
	double tmp;
	if (x <= -0.056) {
		tmp = 1.0 / (fma(t_3, 1.5, 3.0) / fma((sin(x) * (sqrt(2.0) * (cos(x) - 1.0))), fma(-0.0625, sin(x), sin(y)), 2.0));
	} else if (x <= 0.23) {
		tmp = fma(fma(t_1, -0.0625, sin(y)), (fma(sin(y), -0.0625, t_1) * t_4), 2.0) / fma(1.5, fma(cos(x), t_2, (cos(y) * t_0)), 3.0);
	} else {
		tmp = (fma(t_4, (-0.0625 * pow(sin(x), 2.0)), 2.0) / ((t_3 / 2.0) + 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = fma(t_2, cos(x), Float64(t_0 * cos(y)))
	t_4 = Float64(Float64(cos(x) - cos(y)) * sqrt(2.0))
	tmp = 0.0
	if (x <= -0.056)
		tmp = Float64(1.0 / Float64(fma(t_3, 1.5, 3.0) / fma(Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))), fma(-0.0625, sin(x), sin(y)), 2.0)));
	elseif (x <= 0.23)
		tmp = Float64(fma(fma(t_1, -0.0625, sin(y)), Float64(fma(sin(y), -0.0625, t_1) * t_4), 2.0) / fma(1.5, fma(cos(x), t_2, Float64(cos(y) * t_0)), 3.0));
	else
		tmp = Float64(Float64(fma(t_4, Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(Float64(t_3 / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.056], N[(1.0 / N[(N[(t$95$3 * 1.5 + 3.0), $MachinePrecision] / N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.23], N[(N[(N[(t$95$1 * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + t$95$1), $MachinePrecision] * t$95$4), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$4 * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$3 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0560000000000000012

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -0.0560000000000000012 < x < 0.23000000000000001

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.4

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.7

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

   11. Applied rewrites 50.7%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. div-flip-rev N/A

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

   13. Applied rewrites 50.7%

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

   if 0.23000000000000001 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.0

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

   6. Applied rewrites 62.0%

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

Alternative 13: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ t_2 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_3 := \mathsf{fma}\left(t\_0, \cos x, t\_2\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_3, 1.5, 3\right)}{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), t\_1, 2\right)}}\\ \mathbf{elif}\;x \leq 530000000:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), t\_1, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, 1, t\_2\right), 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{\frac{t\_3}{2} + 1}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (fma -0.0625 (sin x) (sin y)))
        (t_2 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_3 (fma t_0 (cos x) t_2)))
   (if (<= x -5e-8)
     (/
      1.0
      (/
       (fma t_3 1.5 3.0)
       (fma (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0))) t_1 2.0)))
     (if (<= x 530000000.0)
       (/
        1.0
        (/
         3.0
         (/
          (fma
           (* (* (- 1.0 (cos y)) (sqrt 2.0)) (fma (sin y) -0.0625 (sin x)))
           t_1
           2.0)
          (fma 0.5 (fma t_0 1.0 t_2) 1.0))))
       (/
        (/
         (fma
          (* (- (cos x) (cos y)) (sqrt 2.0))
          (* -0.0625 (pow (sin x) 2.0))
          2.0)
         (+ (/ t_3 2.0) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma(-0.0625, sin(x), sin(y));
	double t_2 = (3.0 - sqrt(5.0)) * cos(y);
	double t_3 = fma(t_0, cos(x), t_2);
	double tmp;
	if (x <= -5e-8) {
		tmp = 1.0 / (fma(t_3, 1.5, 3.0) / fma((sin(x) * (sqrt(2.0) * (cos(x) - 1.0))), t_1, 2.0));
	} else if (x <= 530000000.0) {
		tmp = 1.0 / (3.0 / (fma((((1.0 - cos(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), t_1, 2.0) / fma(0.5, fma(t_0, 1.0, t_2), 1.0)));
	} else {
		tmp = (fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * pow(sin(x), 2.0)), 2.0) / ((t_3 / 2.0) + 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(-0.0625, sin(x), sin(y))
	t_2 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_3 = fma(t_0, cos(x), t_2)
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(1.0 / Float64(fma(t_3, 1.5, 3.0) / fma(Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))), t_1, 2.0)));
	elseif (x <= 530000000.0)
		tmp = Float64(1.0 / Float64(3.0 / Float64(fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), t_1, 2.0) / fma(0.5, fma(t_0, 1.0, t_2), 1.0))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(Float64(t_3 / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $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[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(1.0 / N[(N[(t$95$3 * 1.5 + 3.0), $MachinePrecision] / N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 530000000.0], N[(1.0 / N[(3.0 / N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * 1.0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$3 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-8

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -4.9999999999999998e-8 < x < 5.3e8

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 59.6%

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

   if 5.3e8 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.0

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

   6. Applied rewrites 62.0%

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

Alternative 14: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_2 := \mathsf{fma}\left(t\_0, \cos x, t\_1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_2, 1.5, 3\right)}{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}}\\ \mathbf{elif}\;x \leq 530000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\frac{\mathsf{fma}\left(t\_0, 1, t\_1\right)}{2} + 1}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{\frac{t\_2}{2} + 1}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_2 (fma t_0 (cos x) t_1)))
   (if (<= x -5e-8)
     (/
      1.0
      (/
       (fma t_2 1.5 3.0)
       (fma
        (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0)))
        (fma -0.0625 (sin x) (sin y))
        2.0)))
     (if (<= x 530000000.0)
       (/
        (/
         (fma
          (* (- 1.0 (cos y)) (sqrt 2.0))
          (* (fma (sin x) -0.0625 (sin y)) (fma (sin y) -0.0625 (sin x)))
          2.0)
         (+ (/ (fma t_0 1.0 t_1) 2.0) 1.0))
        3.0)
       (/
        (/
         (fma
          (* (- (cos x) (cos y)) (sqrt 2.0))
          (* -0.0625 (pow (sin x) 2.0))
          2.0)
         (+ (/ t_2 2.0) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (3.0 - sqrt(5.0)) * cos(y);
	double t_2 = fma(t_0, cos(x), t_1);
	double tmp;
	if (x <= -5e-8) {
		tmp = 1.0 / (fma(t_2, 1.5, 3.0) / fma((sin(x) * (sqrt(2.0) * (cos(x) - 1.0))), fma(-0.0625, sin(x), sin(y)), 2.0));
	} else if (x <= 530000000.0) {
		tmp = (fma(((1.0 - cos(y)) * sqrt(2.0)), (fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / ((fma(t_0, 1.0, t_1) / 2.0) + 1.0)) / 3.0;
	} else {
		tmp = (fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * pow(sin(x), 2.0)), 2.0) / ((t_2 / 2.0) + 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_2 = fma(t_0, cos(x), t_1)
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(1.0 / Float64(fma(t_2, 1.5, 3.0) / fma(Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))), fma(-0.0625, sin(x), sin(y)), 2.0)));
	elseif (x <= 530000000.0)
		tmp = Float64(Float64(fma(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / Float64(Float64(fma(t_0, 1.0, t_1) / 2.0) + 1.0)) / 3.0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(Float64(t_2 / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(1.0 / N[(N[(t$95$2 * 1.5 + 3.0), $MachinePrecision] / N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 530000000.0], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$2 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-8

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -4.9999999999999998e-8 < x < 5.3e8

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 62.3%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 59.6%

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

   if 5.3e8 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.0

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

   6. Applied rewrites 62.0%

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

Alternative 15: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ t_2 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_3 := \mathsf{fma}\left(t\_0, \cos x, t\_2\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_3, 1.5, 3\right)}{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), t\_1, 2\right)}}\\ \mathbf{elif}\;x \leq 530000000:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 1, t\_2\right), 1.5, 3\right)}{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), t\_1, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{\frac{t\_3}{2} + 1}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (fma -0.0625 (sin x) (sin y)))
        (t_2 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_3 (fma t_0 (cos x) t_2)))
   (if (<= x -5e-8)
     (/
      1.0
      (/
       (fma t_3 1.5 3.0)
       (fma (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0))) t_1 2.0)))
     (if (<= x 530000000.0)
       (/
        1.0
        (/
         (fma (fma t_0 1.0 t_2) 1.5 3.0)
         (fma
          (* (* (- 1.0 (cos y)) (sqrt 2.0)) (fma (sin y) -0.0625 (sin x)))
          t_1
          2.0)))
       (/
        (/
         (fma
          (* (- (cos x) (cos y)) (sqrt 2.0))
          (* -0.0625 (pow (sin x) 2.0))
          2.0)
         (+ (/ t_3 2.0) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma(-0.0625, sin(x), sin(y));
	double t_2 = (3.0 - sqrt(5.0)) * cos(y);
	double t_3 = fma(t_0, cos(x), t_2);
	double tmp;
	if (x <= -5e-8) {
		tmp = 1.0 / (fma(t_3, 1.5, 3.0) / fma((sin(x) * (sqrt(2.0) * (cos(x) - 1.0))), t_1, 2.0));
	} else if (x <= 530000000.0) {
		tmp = 1.0 / (fma(fma(t_0, 1.0, t_2), 1.5, 3.0) / fma((((1.0 - cos(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), t_1, 2.0));
	} else {
		tmp = (fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * pow(sin(x), 2.0)), 2.0) / ((t_3 / 2.0) + 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(-0.0625, sin(x), sin(y))
	t_2 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_3 = fma(t_0, cos(x), t_2)
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(1.0 / Float64(fma(t_3, 1.5, 3.0) / fma(Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))), t_1, 2.0)));
	elseif (x <= 530000000.0)
		tmp = Float64(1.0 / Float64(fma(fma(t_0, 1.0, t_2), 1.5, 3.0) / fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), t_1, 2.0)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(Float64(t_3 / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $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[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(1.0 / N[(N[(t$95$3 * 1.5 + 3.0), $MachinePrecision] / N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 530000000.0], N[(1.0 / N[(N[(N[(t$95$0 * 1.0 + t$95$2), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision] / N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$3 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-8

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -4.9999999999999998e-8 < x < 5.3e8

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 59.6%

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

   if 5.3e8 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.0

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

   6. Applied rewrites 62.0%

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

Alternative 16: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(t\_1, \cos x, t\_0\right)\\ t_3 := x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_2, 1.5, 3\right)}{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}}\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, 1, t\_0\right), 1.5, 3\right)}{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, t\_3\right), \mathsf{fma}\left(-0.0625, t\_3, \sin y\right), 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin x}^{2}, 2\right)}{\frac{t\_2}{2} + 1}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (fma t_1 (cos x) t_0))
        (t_3 (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0))))))
   (if (<= x -5e-8)
     (/
      1.0
      (/
       (fma t_2 1.5 3.0)
       (fma
        (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0)))
        (fma -0.0625 (sin x) (sin y))
        2.0)))
     (if (<= x 0.22)
       (/
        1.0
        (/
         (fma (fma t_1 1.0 t_0) 1.5 3.0)
         (fma
          (* (* (- 1.0 (cos y)) (sqrt 2.0)) (fma (sin y) -0.0625 t_3))
          (fma -0.0625 t_3 (sin y))
          2.0)))
       (/
        (/
         (fma
          (* (- (cos x) (cos y)) (sqrt 2.0))
          (* -0.0625 (pow (sin x) 2.0))
          2.0)
         (+ (/ t_2 2.0) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) * cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma(t_1, cos(x), t_0);
	double t_3 = x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)));
	double tmp;
	if (x <= -5e-8) {
		tmp = 1.0 / (fma(t_2, 1.5, 3.0) / fma((sin(x) * (sqrt(2.0) * (cos(x) - 1.0))), fma(-0.0625, sin(x), sin(y)), 2.0));
	} else if (x <= 0.22) {
		tmp = 1.0 / (fma(fma(t_1, 1.0, t_0), 1.5, 3.0) / fma((((1.0 - cos(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, t_3)), fma(-0.0625, t_3, sin(y)), 2.0));
	} else {
		tmp = (fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * pow(sin(x), 2.0)), 2.0) / ((t_2 / 2.0) + 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(t_1, cos(x), t_0)
	t_3 = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))))
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(1.0 / Float64(fma(t_2, 1.5, 3.0) / fma(Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))), fma(-0.0625, sin(x), sin(y)), 2.0)));
	elseif (x <= 0.22)
		tmp = Float64(1.0 / Float64(fma(fma(t_1, 1.0, t_0), 1.5, 3.0) / fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, t_3)), fma(-0.0625, t_3, sin(y)), 2.0)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * (sin(x) ^ 2.0)), 2.0) / Float64(Float64(t_2 / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(1.0 / N[(N[(t$95$2 * 1.5 + 3.0), $MachinePrecision] / N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.22], N[(1.0 / N[(N[(N[(t$95$1 * 1.0 + t$95$0), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision] / N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + t$95$3), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$3 + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$2 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-8

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -4.9999999999999998e-8 < x < 0.220000000000000001

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.4

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.7

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

   11. Applied rewrites 50.7%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 50.3%

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

   15. Taylor expanded in x around 0

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

   17. Applied rewrites 50.8%

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

   if 0.220000000000000001 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.0

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

   6. Applied rewrites 62.0%

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

Alternative 17: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin x}^{2}\\ t_2 := \cos x - \cos y\\ t_3 := 3 - \sqrt{5}\\ t_4 := t\_3 \cdot \cos y\\ t_5 := x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot t\_2}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{t\_3}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 1, t\_4\right), 1.5, 3\right)}{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, t\_5\right), \mathsf{fma}\left(-0.0625, t\_5, \sin y\right), 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, -0.0625 \cdot t\_1, 2\right)}{\frac{\mathsf{fma}\left(t\_0, \cos x, t\_4\right)}{2} + 1}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (* t_3 (cos y)))
        (t_5 (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0))))))
   (if (<= x -5e-8)
     (/
      (+ 2.0 (* (* -0.0625 (* t_1 (sqrt 2.0))) t_2))
      (* 3.0 (+ (+ 1.0 (* (/ t_0 2.0) (cos x))) (* (/ t_3 2.0) (cos y)))))
     (if (<= x 0.22)
       (/
        1.0
        (/
         (fma (fma t_0 1.0 t_4) 1.5 3.0)
         (fma
          (* (* (- 1.0 (cos y)) (sqrt 2.0)) (fma (sin y) -0.0625 t_5))
          (fma -0.0625 t_5 (sin y))
          2.0)))
       (/
        (/
         (fma (* t_2 (sqrt 2.0)) (* -0.0625 t_1) 2.0)
         (+ (/ (fma t_0 (cos x) t_4) 2.0) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = pow(sin(x), 2.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 * cos(y);
	double t_5 = x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)));
	double tmp;
	if (x <= -5e-8) {
		tmp = (2.0 + ((-0.0625 * (t_1 * sqrt(2.0))) * t_2)) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + ((t_3 / 2.0) * cos(y))));
	} else if (x <= 0.22) {
		tmp = 1.0 / (fma(fma(t_0, 1.0, t_4), 1.5, 3.0) / fma((((1.0 - cos(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, t_5)), fma(-0.0625, t_5, sin(y)), 2.0));
	} else {
		tmp = (fma((t_2 * sqrt(2.0)), (-0.0625 * t_1), 2.0) / ((fma(t_0, cos(x), t_4) / 2.0) + 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 * cos(y))
	t_5 = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))))
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_1 * sqrt(2.0))) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(t_3 / 2.0) * cos(y)))));
	elseif (x <= 0.22)
		tmp = Float64(1.0 / Float64(fma(fma(t_0, 1.0, t_4), 1.5, 3.0) / fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, t_5)), fma(-0.0625, t_5, sin(y)), 2.0)));
	else
		tmp = Float64(Float64(fma(Float64(t_2 * sqrt(2.0)), Float64(-0.0625 * t_1), 2.0) / Float64(Float64(fma(t_0, cos(x), t_4) / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $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$3 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.22], N[(1.0 / N[(N[(N[(t$95$0 * 1.0 + t$95$4), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision] / N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + t$95$5), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$5 + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$4), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.0

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

   4. Applied rewrites 62.0%

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

   if -4.9999999999999998e-8 < x < 0.220000000000000001

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.4

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.7

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

   11. Applied rewrites 50.7%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 50.3%

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

   15. Taylor expanded in x around 0

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

   17. Applied rewrites 50.8%

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

   if 0.220000000000000001 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.0

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

   6. Applied rewrites 62.0%

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

Alternative 18: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{1}{\frac{3}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right), 1\right)}}}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          1.0
          (/
           3.0
           (/
            (+
             2.0
             (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
            (fma 0.5 (fma t_0 (cos x) (* t_1 (cos y))) 1.0))))))
   (if (<= y -4.4e-6)
     t_2
     (if (<= y 3.9e-10)
       (*
        (/
         (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)
       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 = 1.0 / (3.0 / ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(0.5, fma(t_0, cos(x), (t_1 * cos(y))), 1.0)));
	double tmp;
	if (y <= -4.4e-6) {
		tmp = t_2;
	} else if (y <= 3.9e-10) {
		tmp = (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;
	} 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(1.0 / Float64(3.0 / Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(0.5, fma(t_0, cos(x), Float64(t_1 * cos(y))), 1.0))))
	tmp = 0.0
	if (y <= -4.4e-6)
		tmp = t_2;
	elseif (y <= 3.9e-10)
		tmp = 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);
	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[(1.0 / N[(3.0 / N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.4e-6], t$95$2, If[LessEqual[y, 3.9e-10], 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], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4000000000000002e-6 or 3.9e-10 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.0

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

   8. Applied rewrites 62.0%

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

   if -4.4000000000000002e-6 < y < 3.9e-10

   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 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.7

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

   6. Applied rewrites 59.7%

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

Alternative 19: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\\ t_2 := \sqrt{5} - 1\\ t_3 := \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right)\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, t\_3, 3\right) \cdot \frac{1}{t\_1}}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, t\_0\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_3, 1.5, 3\right)}{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (+
          2.0
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y)))))))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (fma t_2 (cos x) (* t_0 (cos y)))))
   (if (<= y -4.4e-6)
     (/ 1.0 (* (fma 1.5 t_3 3.0) (/ 1.0 t_1)))
     (if (<= y 3.9e-10)
       (*
        (/
         (fma
          (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x))))
          2.0)
         (fma (fma t_2 (cos x) t_0) 0.5 1.0))
        0.3333333333333333)
       (/ 1.0 (/ (fma t_3 1.5 3.0) t_1))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))));
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = fma(t_2, cos(x), (t_0 * cos(y)));
	double tmp;
	if (y <= -4.4e-6) {
		tmp = 1.0 / (fma(1.5, t_3, 3.0) * (1.0 / t_1));
	} else if (y <= 3.9e-10) {
		tmp = (fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) / fma(fma(t_2, cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333;
	} else {
		tmp = 1.0 / (fma(t_3, 1.5, 3.0) / t_1);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = fma(t_2, cos(x), Float64(t_0 * cos(y)))
	tmp = 0.0
	if (y <= -4.4e-6)
		tmp = Float64(1.0 / Float64(fma(1.5, t_3, 3.0) * Float64(1.0 / t_1)));
	elseif (y <= 3.9e-10)
		tmp = 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_2, cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / Float64(fma(t_3, 1.5, 3.0) / 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[(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]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.4e-6], N[(1.0 / N[(N[(1.5 * t$95$3 + 3.0), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e-10], 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$2 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / N[(N[(t$95$3 * 1.5 + 3.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4000000000000002e-6

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.0

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

   10. Applied rewrites 62.0%

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

   if -4.4000000000000002e-6 < y < 3.9e-10

   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 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.7

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

   6. Applied rewrites 59.7%

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

   if 3.9e-10 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.1

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

   8. Applied rewrites 62.1%

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

Alternative 20: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_0 \cdot \cos y\right), 1.5, 3\right)}{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_0\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          1.0
          (/
           (fma (fma t_1 (cos x) (* t_0 (cos y))) 1.5 3.0)
           (+
            2.0
            (*
             -0.0625
             (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))))))
   (if (<= y -4.4e-6)
     t_2
     (if (<= y 3.9e-10)
       (*
        (/
         (fma
          (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x))))
          2.0)
         (fma (fma t_1 (cos x) t_0) 0.5 1.0))
        0.3333333333333333)
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 1.0 / (fma(fma(t_1, cos(x), (t_0 * cos(y))), 1.5, 3.0) / (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))));
	double tmp;
	if (y <= -4.4e-6) {
		tmp = t_2;
	} else if (y <= 3.9e-10) {
		tmp = (fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) / fma(fma(t_1, cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(1.0 / Float64(fma(fma(t_1, cos(x), Float64(t_0 * cos(y))), 1.5, 3.0) / Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))))))
	tmp = 0.0
	if (y <= -4.4e-6)
		tmp = t_2;
	elseif (y <= 3.9e-10)
		tmp = 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_1, cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision] / 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]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.4e-6], t$95$2, If[LessEqual[y, 3.9e-10], 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$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4000000000000002e-6 or 3.9e-10 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -4.4000000000000002e-6 < y < 3.9e-10

   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 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.7

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

   6. Applied rewrites 59.7%

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

Alternative 21: 78.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1\right), 0.5, 1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{t\_2}{t\_0 \cdot 0.3333333333333333}}\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + 0.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_1\right) - 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

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 (fma (- (sqrt 5.0) 1.0) (cos x) t_1) 0.5 1.0)))
   (if (<= x -5e-8)
     (/ 1.0 (/ t_2 (* t_0 0.3333333333333333)))
     (if (<= x 0.22)
       (/
        1.0
        (/
         3.0
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
          (+ 1.0 (* 0.5 (- (+ (sqrt 5.0) (* (cos y) t_1)) 1.0))))))
       (* (/ t_0 t_2) 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(fma((sqrt(5.0) - 1.0), cos(x), t_1), 0.5, 1.0);
	double tmp;
	if (x <= -5e-8) {
		tmp = 1.0 / (t_2 / (t_0 * 0.3333333333333333));
	} else if (x <= 0.22) {
		tmp = 1.0 / (3.0 / ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (1.0 + (0.5 * ((sqrt(5.0) + (cos(y) * t_1)) - 1.0)))));
	} else {
		tmp = (t_0 / t_2) * 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(fma(Float64(sqrt(5.0) - 1.0), cos(x), t_1), 0.5, 1.0)
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(1.0 / Float64(t_2 / Float64(t_0 * 0.3333333333333333)));
	elseif (x <= 0.22)
		tmp = Float64(1.0 / Float64(3.0 / Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(1.0 + Float64(0.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * t_1)) - 1.0))))));
	else
		tmp = Float64(Float64(t_0 / t_2) * 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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(1.0 / N[(t$95$2 / N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.22], N[(1.0 / N[(3.0 / N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-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 59.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   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 59.7%

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

   if -4.9999999999999998e-8 < x < 0.220000000000000001

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.1%

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

   if 0.220000000000000001 < 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 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.7

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

   6. Applied rewrites 59.7%

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

Alternative 22: 78.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1\right), 0.5, 1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{t\_2}{t\_0 \cdot 0.3333333333333333}}\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{1}{3 \cdot \frac{1 + 0.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_1\right) - 1\right)}{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

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 (fma (- (sqrt 5.0) 1.0) (cos x) t_1) 0.5 1.0)))
   (if (<= x -5e-8)
     (/ 1.0 (/ t_2 (* t_0 0.3333333333333333)))
     (if (<= x 0.22)
       (/
        1.0
        (*
         3.0
         (/
          (+ 1.0 (* 0.5 (- (+ (sqrt 5.0) (* (cos y) t_1)) 1.0)))
          (+
           2.0
           (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y)))))))))
       (* (/ t_0 t_2) 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(fma((sqrt(5.0) - 1.0), cos(x), t_1), 0.5, 1.0);
	double tmp;
	if (x <= -5e-8) {
		tmp = 1.0 / (t_2 / (t_0 * 0.3333333333333333));
	} else if (x <= 0.22) {
		tmp = 1.0 / (3.0 * ((1.0 + (0.5 * ((sqrt(5.0) + (cos(y) * t_1)) - 1.0))) / (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y))))))));
	} else {
		tmp = (t_0 / t_2) * 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(fma(Float64(sqrt(5.0) - 1.0), cos(x), t_1), 0.5, 1.0)
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(1.0 / Float64(t_2 / Float64(t_0 * 0.3333333333333333)));
	elseif (x <= 0.22)
		tmp = Float64(1.0 / Float64(3.0 * Float64(Float64(1.0 + Float64(0.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * t_1)) - 1.0))) / Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))))));
	else
		tmp = Float64(Float64(t_0 / t_2) * 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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(1.0 / N[(t$95$2 / N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.22], N[(1.0 / N[(3.0 * N[(N[(1.0 + N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-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 59.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   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 59.7%

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

   if -4.9999999999999998e-8 < x < 0.220000000000000001

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \frac{\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2}}, \frac{0.6666666666666666}{\frac{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}\right)} \]

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 59.1%

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

   if 0.220000000000000001 < 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 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.7

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

   6. Applied rewrites 59.7%

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

Alternative 23: 78.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1\right), 0.5, 1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{t\_2}{t\_0 \cdot 0.3333333333333333}}\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + 0.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_1\right) - 1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

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 (fma (- (sqrt 5.0) 1.0) (cos x) t_1) 0.5 1.0)))
   (if (<= x -5e-8)
     (/ 1.0 (/ t_2 (* t_0 0.3333333333333333)))
     (if (<= x 0.22)
       (/
        (/
         (+
          2.0
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
         (+ 1.0 (* 0.5 (- (+ (sqrt 5.0) (* (cos y) t_1)) 1.0))))
        3.0)
       (* (/ t_0 t_2) 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(fma((sqrt(5.0) - 1.0), cos(x), t_1), 0.5, 1.0);
	double tmp;
	if (x <= -5e-8) {
		tmp = 1.0 / (t_2 / (t_0 * 0.3333333333333333));
	} else if (x <= 0.22) {
		tmp = ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (1.0 + (0.5 * ((sqrt(5.0) + (cos(y) * t_1)) - 1.0)))) / 3.0;
	} else {
		tmp = (t_0 / t_2) * 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(fma(Float64(sqrt(5.0) - 1.0), cos(x), t_1), 0.5, 1.0)
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(1.0 / Float64(t_2 / Float64(t_0 * 0.3333333333333333)));
	elseif (x <= 0.22)
		tmp = Float64(Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(1.0 + Float64(0.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * t_1)) - 1.0)))) / 3.0);
	else
		tmp = Float64(Float64(t_0 / t_2) * 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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(1.0 / N[(t$95$2 / N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.22], N[(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[(1.0 + N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(t$95$0 / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-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 59.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   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 59.7%

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

   if -4.9999999999999998e-8 < x < 0.220000000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 59.1%

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

   if 0.220000000000000001 < 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 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.7

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

   6. Applied rewrites 59.7%

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

Alternative 24: 78.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1\right), 0.5, 1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{t\_2}{t\_0 \cdot 0.3333333333333333}}\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{1 + 0.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_1\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

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 (fma (- (sqrt 5.0) 1.0) (cos x) t_1) 0.5 1.0)))
   (if (<= x -5e-8)
     (/ 1.0 (/ t_2 (* t_0 0.3333333333333333)))
     (if (<= x 0.22)
       (*
        0.3333333333333333
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0)
         (+ 1.0 (* 0.5 (- (+ (sqrt 5.0) (* (cos y) t_1)) 1.0)))))
       (* (/ t_0 t_2) 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(fma((sqrt(5.0) - 1.0), cos(x), t_1), 0.5, 1.0);
	double tmp;
	if (x <= -5e-8) {
		tmp = 1.0 / (t_2 / (t_0 * 0.3333333333333333));
	} else if (x <= 0.22) {
		tmp = 0.3333333333333333 * (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (1.0 + (0.5 * ((sqrt(5.0) + (cos(y) * t_1)) - 1.0))));
	} else {
		tmp = (t_0 / t_2) * 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(fma(Float64(sqrt(5.0) - 1.0), cos(x), t_1), 0.5, 1.0)
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(1.0 / Float64(t_2 / Float64(t_0 * 0.3333333333333333)));
	elseif (x <= 0.22)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(1.0 + Float64(0.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * t_1)) - 1.0)))));
	else
		tmp = Float64(Float64(t_0 / t_2) * 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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[x, -5e-8], N[(1.0 / N[(t$95$2 / N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.22], N[(0.3333333333333333 * N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e-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 59.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   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 59.7%

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

   if -4.9999999999999998e-8 < x < 0.220000000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 59.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 59.0%

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

   if 0.220000000000000001 < 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 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.7

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

   6. Applied rewrites 59.7%

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

Alternative 25: 78.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{1 + 0.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_0\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
            (- 0.5 (* 0.5 (cos (* 2.0 x))))
            2.0)
           (fma (fma (- (sqrt 5.0) 1.0) (cos x) t_0) 0.5 1.0))
          0.3333333333333333)))
   (if (<= x -5e-8)
     t_1
     (if (<= x 0.22)
       (*
        0.3333333333333333
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0)
         (+ 1.0 (* 0.5 (- (+ (sqrt 5.0) (* (cos y) t_0)) 1.0)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -5e-8) {
		tmp = t_1;
	} else if (x <= 0.22) {
		tmp = 0.3333333333333333 * (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (1.0 + (0.5 * ((sqrt(5.0) + (cos(y) * t_0)) - 1.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = 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(Float64(sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -5e-8)
		tmp = t_1;
	elseif (x <= 0.22)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(1.0 + Float64(0.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * t_0)) - 1.0)))));
	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[(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[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -5e-8], t$95$1, If[LessEqual[x, 0.22], N[(0.3333333333333333 * N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-8 or 0.220000000000000001 < 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 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.7

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

   6. Applied rewrites 59.7%

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

   if -4.9999999999999998e-8 < x < 0.220000000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 59.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 59.0%

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

Alternative 26: 78.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.0625, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right) - 1, -0.5, -1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
            (- 0.5 (* 0.5 (cos (* 2.0 x))))
            2.0)
           (fma (fma (- (sqrt 5.0) 1.0) (cos x) t_0) 0.5 1.0))
          0.3333333333333333)))
   (if (<= x -5e-8)
     t_1
     (if (<= x 0.22)
       (/
        (/
         (fma
          0.0625
          (* (- 0.5 (* 0.5 (cos (* 2.0 y)))) (* (- 1.0 (cos y)) (sqrt 2.0)))
          -2.0)
         (fma (- (fma (cos y) t_0 (sqrt 5.0)) 1.0) -0.5 -1.0))
        3.0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -5e-8) {
		tmp = t_1;
	} else if (x <= 0.22) {
		tmp = (fma(0.0625, ((0.5 - (0.5 * cos((2.0 * y)))) * ((1.0 - cos(y)) * sqrt(2.0))), -2.0) / fma((fma(cos(y), t_0, sqrt(5.0)) - 1.0), -0.5, -1.0)) / 3.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = 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(Float64(sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -5e-8)
		tmp = t_1;
	elseif (x <= 0.22)
		tmp = Float64(Float64(fma(0.0625, Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y)))) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))), -2.0) / fma(Float64(fma(cos(y), t_0, sqrt(5.0)) - 1.0), -0.5, -1.0)) / 3.0);
	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[(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[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -5e-8], t$95$1, If[LessEqual[x, 0.22], N[(N[(N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-8 or 0.220000000000000001 < 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 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.7

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

   6. Applied rewrites 59.7%

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

   if -4.9999999999999998e-8 < x < 0.220000000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 59.1%

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

   8. Applied rewrites 59.1%

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

Alternative 27: 78.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.22:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right) - 1, -0.5, -1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
            (- 0.5 (* 0.5 (cos (* 2.0 x))))
            2.0)
           (fma (fma (- (sqrt 5.0) 1.0) (cos x) t_0) 0.5 1.0))
          0.3333333333333333)))
   (if (<= x -5e-8)
     t_1
     (if (<= x 0.22)
       (*
        (/
         (fma
          0.0625
          (* (- 0.5 (* 0.5 (cos (* 2.0 y)))) (* (- 1.0 (cos y)) (sqrt 2.0)))
          -2.0)
         (fma (- (fma (cos y) t_0 (sqrt 5.0)) 1.0) -0.5 -1.0))
        0.3333333333333333)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -5e-8) {
		tmp = t_1;
	} else if (x <= 0.22) {
		tmp = (fma(0.0625, ((0.5 - (0.5 * cos((2.0 * y)))) * ((1.0 - cos(y)) * sqrt(2.0))), -2.0) / fma((fma(cos(y), t_0, sqrt(5.0)) - 1.0), -0.5, -1.0)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = 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(Float64(sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -5e-8)
		tmp = t_1;
	elseif (x <= 0.22)
		tmp = Float64(Float64(fma(0.0625, Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y)))) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))), -2.0) / fma(Float64(fma(cos(y), t_0, sqrt(5.0)) - 1.0), -0.5, -1.0)) * 0.3333333333333333);
	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[(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[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -5e-8], t$95$1, If[LessEqual[x, 0.22], N[(N[(N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-8 or 0.220000000000000001 < 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 59.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.7

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

   6. Applied rewrites 59.7%

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

   if -4.9999999999999998e-8 < x < 0.220000000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 59.1%

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

   8. Applied rewrites 59.0%

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

Alternative 28: 59.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. 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 59.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 59.7

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

6. Applied rewrites 59.7%

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

Alternative 29: 42.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) + 1.5\right) - 0.5 \cdot \sqrt{5}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   2.0
   (-
    (+ (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 1.5)
    (* 0.5 (sqrt 5.0))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * (2.0 / ((fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) + 1.5) - (0.5 * sqrt(5.0))));
}

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(2.0 / Float64(Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) + 1.5) - Float64(0.5 * sqrt(5.0)))))
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(2.0 / N[(N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] + 1.5), $MachinePrecision] - N[(0.5 * N[Sqrt[5.0], $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 59.7%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-+r+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lift--.f64 N/A

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

   9. div-sub N/A

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

   10. mult-flip-rev N/A

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

   11. metadata-eval N/A

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

   12. associate-+r- N/A

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

   13. lower--.f64 N/A

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

6. Applied rewrites 59.6%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) + 1.5\right) - \color{blue}{0.5 \cdot \sqrt{5}}} \]

7. Taylor expanded in x around 0

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

9. Applied rewrites 42.6%

   \[\leadsto 0.3333333333333333 \cdot \frac{2}{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) + 1.5\right)} - 0.5 \cdot \sqrt{5}} \]
10. Add Preprocessing

Alternative 30: 42.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2}{1 + 0.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)} \end{array} \]

FPCore

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

C

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

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 = 0.3333333333333333d0 * (2.0d0 / (1.0d0 + (0.5d0 * ((sqrt(5.0d0) + (cos(y) * (3.0d0 - sqrt(5.0d0)))) - 1.0d0))))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * (2.0 / (1.0 + (0.5 * ((Math.sqrt(5.0) + (Math.cos(y) * (3.0 - Math.sqrt(5.0)))) - 1.0))));
}

Python

def code(x, y):
	return 0.3333333333333333 * (2.0 / (1.0 + (0.5 * ((math.sqrt(5.0) + (math.cos(y) * (3.0 - math.sqrt(5.0)))) - 1.0))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * (2.0 / (1.0 + (0.5 * ((sqrt(5.0) + (cos(y) * (3.0 - sqrt(5.0)))) - 1.0))));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

6. Applied rewrites 59.1%

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

7. Taylor expanded in y around 0

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

9. Applied rewrites 42.0%

   \[\leadsto 0.3333333333333333 \cdot \frac{2}{\color{blue}{1} + 0.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)} \]
10. Add Preprocessing

Alternative 31: 40.1% accurate, 316.7× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. 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 59.7%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-+r+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lift--.f64 N/A

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

   9. div-sub N/A

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

   10. mult-flip-rev N/A

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

   11. metadata-eval N/A

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

   12. associate-+r- N/A

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

   13. lower--.f64 N/A

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

6. Applied rewrites 59.6%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) + 1.5\right) - \color{blue}{0.5 \cdot \sqrt{5}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
8. Step-by-step derivation

9. Applied rewrites 40.1%

   \[\leadsto 0.3333333333333333 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025155 
(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))))))