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

Percentage Accurate: 99.3% → 99.4%

Time: 12.4s

Alternatives: 34

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 1\right), \frac{-1}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y, 1.5, -3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}, \frac{0.6666666666666666}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, 2\right)\right)}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma
  (fma
   (* (- (cos x) (cos y)) (sqrt 2.0))
   (* (fma (sin y) -0.0625 (sin x)) (fma -0.0625 (sin x) (sin y)))
   1.0)
  (/
   -1.0
   (fma
    (* (- (sqrt 5.0) 3.0) (cos y))
    1.5
    (* -3.0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
  (/
   0.6666666666666666
   (fma (- (sqrt 5.0) 1.0) (cos x) (fma (- 3.0 (sqrt 5.0)) (cos y) 2.0)))))

C

double code(double x, double y) {
	return fma(fma(((cos(x) - cos(y)) * sqrt(2.0)), (fma(sin(y), -0.0625, sin(x)) * fma(-0.0625, sin(x), sin(y))), 1.0), (-1.0 / fma(((sqrt(5.0) - 3.0) * cos(y)), 1.5, (-3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)))), (0.6666666666666666 / fma((sqrt(5.0) - 1.0), cos(x), fma((3.0 - sqrt(5.0)), cos(y), 2.0))));
}

Julia

function code(x, y)
	return fma(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(fma(sin(y), -0.0625, sin(x)) * fma(-0.0625, sin(x), sin(y))), 1.0), Float64(-1.0 / fma(Float64(Float64(sqrt(5.0) - 3.0) * cos(y)), 1.5, Float64(-3.0 * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)))), Float64(0.6666666666666666 / fma(Float64(sqrt(5.0) - 1.0), cos(x), fma(Float64(3.0 - sqrt(5.0)), cos(y), 2.0))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-1.0 / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(-3.0 * N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / 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] + 2.0), $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%

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

5. Applied rewrites 99.3%

   \[\leadsto \mathsf{fma}\left(\frac{2}{\left(2 + \left(\sqrt{5} - 1\right) \cdot \cos x\right) + \left(3 - \sqrt{5}\right) \cdot \cos y}, 0.3333333333333333, \frac{1 - \left(\cos y - \cos x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3 - \left(\sqrt{5} - 3\right) \cdot \left(\cos y \cdot 1.5\right)}}\right) \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. add-flip N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-+r- N/A

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

   11. lower--.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

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

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

   16. lower-*.f64 N/A

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

7. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\frac{2}{\color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, 2\right) - \left(1 - \sqrt{5}\right) \cdot \cos x}}, 0.3333333333333333, \frac{1 - \left(\cos y - \cos x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3 - \left(\sqrt{5} - 3\right) \cdot \left(\cos y \cdot 1.5\right)}\right) \]

9. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 1\right), \frac{-1}{\mathsf{fma}\left(\left(\sqrt{5} - 3\right) \cdot \cos y, 1.5, -3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}, \frac{0.6666666666666666}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, 2\right)\right)}\right)} \]
10. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(1.0 - Float64(Float64(sqrt(5.0) - 3.0) * Float64(0.5 * cos(y)))))) / 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[(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[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(1.0 - N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $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%

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

5. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(Float64(3.0 - sqrt(5.0)), Float64(0.5 * cos(y)), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))) / 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[(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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $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%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. mult-flip-rev N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. div-add-rev N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l/ N/A

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

   14. lift-/.f64 N/A

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

5. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

5. Applied rewrites 99.3%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 99.3%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0)) / 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[(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[(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] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $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%

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

Alternative 7: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \end{array} \]

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)))
   0.3333333333333333
   0.6666666666666666)
  (fma
   0.5
   (fma (cos y) (- 3.0 (sqrt 5.0)) (* (cos x) (- (sqrt 5.0) 1.0)))
   1.0)))

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))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), (cos(x) * (sqrt(5.0) - 1.0))), 1.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))) * fma(-0.0625, sin(x), sin(y))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(cos(x) * Float64(sqrt(5.0) - 1.0))), 1.0))
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]), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 0.3333333333333333, 0.6666666666666666\right)}}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]
8. Add Preprocessing

Alternative 8: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.2%

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

Alternative 9: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0)) * 0.3333333333333333)
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[(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[(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] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.2%

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

Alternative 10: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := \left(t\_1 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \sin y, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_0, \cos x, t\_3 \cdot \cos y\right)}{2}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \left(3 + \cos x \cdot t\_0\right) - \sqrt{5}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_1}{3 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) + \frac{t\_3}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (* (* t_1 (sqrt 2.0)) (fma (sin y) -0.0625 (sin x))))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= y -6.5e-8)
     (/
      (fma t_2 (sin y) 2.0)
      (+ 3.0 (* 3.0 (/ (fma t_0 (cos x) (* t_3 (cos y))) 2.0))))
     (if (<= y 1.75e-7)
       (/
        (* (fma t_2 (fma -0.0625 (sin x) (sin y)) 2.0) 0.3333333333333333)
        (fma 0.5 (- (+ 3.0 (* (cos x) t_0)) (sqrt 5.0)) 1.0))
       (/
        (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_1))
        (*
         3.0
         (+
          (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
          (* (/ t_3 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = (t_1 * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x));
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -6.5e-8) {
		tmp = fma(t_2, sin(y), 2.0) / (3.0 + (3.0 * (fma(t_0, cos(x), (t_3 * cos(y))) / 2.0)));
	} else if (y <= 1.75e-7) {
		tmp = (fma(t_2, fma(-0.0625, sin(x), sin(y)), 2.0) * 0.3333333333333333) / fma(0.5, ((3.0 + (cos(x) * t_0)) - sqrt(5.0)), 1.0);
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_1)) / (3.0 * (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) + ((t_3 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(Float64(t_1 * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x)))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -6.5e-8)
		tmp = Float64(fma(t_2, sin(y), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_0, cos(x), Float64(t_3 * cos(y))) / 2.0))));
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(fma(t_2, fma(-0.0625, sin(x), sin(y)), 2.0) * 0.3333333333333333) / fma(0.5, Float64(Float64(3.0 + Float64(cos(x) * t_0)) - sqrt(5.0)), 1.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_1)) / Float64(3.0 * Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0) + Float64(Float64(t_3 / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e-8], N[(N[(t$95$2 * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-7], N[(N[(N[(t$95$2 * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.49999999999999997e-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 x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 64.5%

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

   8. Applied rewrites 64.5%

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

   if -6.49999999999999997e-8 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 60.0

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

   8. Applied rewrites 60.0%

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

   if 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 64.5

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. div-add N/A

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

      9. mult-flip-rev N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 64.5

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

   6. Applied rewrites 64.5%

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

Alternative 11: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ t_3 := \left(t\_2 \cdot \sqrt{2}\right) \cdot t\_0\\ t_4 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, \sin y, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, \cos x, t\_4 \cdot \cos y\right)}{2}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \left(3 + \cos x \cdot t\_1\right) - \sqrt{5}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t\_0 \cdot \sqrt{2}\right) \cdot \left(t\_2 \cdot \sin y\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_4}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (* (* t_2 (sqrt 2.0)) t_0))
        (t_4 (- 3.0 (sqrt 5.0))))
   (if (<= y -6.5e-8)
     (/
      (fma t_3 (sin y) 2.0)
      (+ 3.0 (* 3.0 (/ (fma t_1 (cos x) (* t_4 (cos y))) 2.0))))
     (if (<= y 1.75e-7)
       (/
        (* (fma t_3 (fma -0.0625 (sin x) (sin y)) 2.0) 0.3333333333333333)
        (fma 0.5 (- (+ 3.0 (* (cos x) t_1)) (sqrt 5.0)) 1.0))
       (/
        (+ 2.0 (* (* t_0 (sqrt 2.0)) (* t_2 (sin y))))
        (*
         3.0
         (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_4 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = (t_2 * sqrt(2.0)) * t_0;
	double t_4 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -6.5e-8) {
		tmp = fma(t_3, sin(y), 2.0) / (3.0 + (3.0 * (fma(t_1, cos(x), (t_4 * cos(y))) / 2.0)));
	} else if (y <= 1.75e-7) {
		tmp = (fma(t_3, fma(-0.0625, sin(x), sin(y)), 2.0) * 0.3333333333333333) / fma(0.5, ((3.0 + (cos(x) * t_1)) - sqrt(5.0)), 1.0);
	} else {
		tmp = (2.0 + ((t_0 * sqrt(2.0)) * (t_2 * sin(y)))) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_4 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(Float64(t_2 * sqrt(2.0)) * t_0)
	t_4 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -6.5e-8)
		tmp = Float64(fma(t_3, sin(y), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, cos(x), Float64(t_4 * cos(y))) / 2.0))));
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(fma(t_3, fma(-0.0625, sin(x), sin(y)), 2.0) * 0.3333333333333333) / fma(0.5, Float64(Float64(3.0 + Float64(cos(x) * t_1)) - sqrt(5.0)), 1.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_0 * sqrt(2.0)) * Float64(t_2 * sin(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_4 / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e-8], N[(N[(t$95$3 * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-7], N[(N[(N[(t$95$3 * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.49999999999999997e-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 x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 64.5%

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

   8. Applied rewrites 64.5%

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

   if -6.49999999999999997e-8 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 60.0

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

   8. Applied rewrites 60.0%

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

   if 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 64.5%

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

Alternative 12: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ t_3 := \left(t\_2 \cdot \sqrt{2}\right) \cdot t\_0\\ t_4 := \cos x \cdot t\_1\\ t_5 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, \sin y, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, \cos x, t\_5 \cdot \cos y\right)}{2}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \left(3 + t\_4\right) - \sqrt{5}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2 \cdot \left(t\_0 \cdot \sqrt{2}\right), \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_5, t\_4\right), 1\right)}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (* (* t_2 (sqrt 2.0)) t_0))
        (t_4 (* (cos x) t_1))
        (t_5 (- 3.0 (sqrt 5.0))))
   (if (<= y -6.5e-8)
     (/
      (fma t_3 (sin y) 2.0)
      (+ 3.0 (* 3.0 (/ (fma t_1 (cos x) (* t_5 (cos y))) 2.0))))
     (if (<= y 1.75e-7)
       (/
        (* (fma t_3 (fma -0.0625 (sin x) (sin y)) 2.0) 0.3333333333333333)
        (fma 0.5 (- (+ 3.0 t_4) (sqrt 5.0)) 1.0))
       (/
        (/
         (fma (* t_2 (* t_0 (sqrt 2.0))) (sin y) 2.0)
         (fma 0.5 (fma (cos y) t_5 t_4) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = (t_2 * sqrt(2.0)) * t_0;
	double t_4 = cos(x) * t_1;
	double t_5 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -6.5e-8) {
		tmp = fma(t_3, sin(y), 2.0) / (3.0 + (3.0 * (fma(t_1, cos(x), (t_5 * cos(y))) / 2.0)));
	} else if (y <= 1.75e-7) {
		tmp = (fma(t_3, fma(-0.0625, sin(x), sin(y)), 2.0) * 0.3333333333333333) / fma(0.5, ((3.0 + t_4) - sqrt(5.0)), 1.0);
	} else {
		tmp = (fma((t_2 * (t_0 * sqrt(2.0))), sin(y), 2.0) / fma(0.5, fma(cos(y), t_5, t_4), 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(Float64(t_2 * sqrt(2.0)) * t_0)
	t_4 = Float64(cos(x) * t_1)
	t_5 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -6.5e-8)
		tmp = Float64(fma(t_3, sin(y), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, cos(x), Float64(t_5 * cos(y))) / 2.0))));
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(fma(t_3, fma(-0.0625, sin(x), sin(y)), 2.0) * 0.3333333333333333) / fma(0.5, Float64(Float64(3.0 + t_4) - sqrt(5.0)), 1.0));
	else
		tmp = Float64(Float64(fma(Float64(t_2 * Float64(t_0 * sqrt(2.0))), sin(y), 2.0) / fma(0.5, fma(cos(y), t_5, t_4), 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e-8], N[(N[(t$95$3 * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$5 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-7], N[(N[(N[(t$95$3 * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(N[(3.0 + t$95$4), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$5 + t$95$4), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.49999999999999997e-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 x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 64.5%

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

   8. Applied rewrites 64.5%

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

   if -6.49999999999999997e-8 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 60.0

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

   8. Applied rewrites 60.0%

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

   if 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

   6. Applied rewrites 64.5%

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

Alternative 13: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x \cdot t\_1\\ t_3 := \cos x - \cos y\\ t_4 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.0054:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_3 \cdot \sqrt{2}\right) \cdot t\_0, \sin y, 2\right)}{3 + 3 \cdot \frac{\mathsf{fma}\left(t\_1, \cos x, t\_4 \cdot \cos y\right)}{2}}\\ \mathbf{elif}\;y \leq 0.00295:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_3}{\mathsf{fma}\left(-0.75, {y}^{2} \cdot t\_4, 3 \cdot \left(1 + \mathsf{fma}\left(0.5, t\_2, 0.5 \cdot t\_4\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_3 \cdot \left(t\_0 \cdot \sqrt{2}\right), \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_4, t\_2\right), 1\right)}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (cos x) t_1))
        (t_3 (- (cos x) (cos y)))
        (t_4 (- 3.0 (sqrt 5.0))))
   (if (<= y -0.0054)
     (/
      (fma (* (* t_3 (sqrt 2.0)) t_0) (sin y) 2.0)
      (+ 3.0 (* 3.0 (/ (fma t_1 (cos x) (* t_4 (cos y))) 2.0))))
     (if (<= y 0.00295)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_3))
        (fma
         -0.75
         (* (pow y 2.0) t_4)
         (* 3.0 (+ 1.0 (fma 0.5 t_2 (* 0.5 t_4))))))
       (/
        (/
         (fma (* t_3 (* t_0 (sqrt 2.0))) (sin y) 2.0)
         (fma 0.5 (fma (cos y) t_4 t_2) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) * t_1;
	double t_3 = cos(x) - cos(y);
	double t_4 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -0.0054) {
		tmp = fma(((t_3 * sqrt(2.0)) * t_0), sin(y), 2.0) / (3.0 + (3.0 * (fma(t_1, cos(x), (t_4 * cos(y))) / 2.0)));
	} else if (y <= 0.00295) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_3)) / fma(-0.75, (pow(y, 2.0) * t_4), (3.0 * (1.0 + fma(0.5, t_2, (0.5 * t_4)))));
	} else {
		tmp = (fma((t_3 * (t_0 * sqrt(2.0))), sin(y), 2.0) / fma(0.5, fma(cos(y), t_4, t_2), 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) * t_1)
	t_3 = Float64(cos(x) - cos(y))
	t_4 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -0.0054)
		tmp = Float64(fma(Float64(Float64(t_3 * sqrt(2.0)) * t_0), sin(y), 2.0) / Float64(3.0 + Float64(3.0 * Float64(fma(t_1, cos(x), Float64(t_4 * cos(y))) / 2.0))));
	elseif (y <= 0.00295)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_3)) / fma(-0.75, Float64((y ^ 2.0) * t_4), Float64(3.0 * Float64(1.0 + fma(0.5, t_2, Float64(0.5 * t_4))))));
	else
		tmp = Float64(Float64(fma(Float64(t_3 * Float64(t_0 * sqrt(2.0))), sin(y), 2.0) / fma(0.5, fma(cos(y), t_4, t_2), 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0054], N[(N[(N[(N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00295], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(-0.75 * N[(N[Power[y, 2.0], $MachinePrecision] * t$95$4), $MachinePrecision] + N[(3.0 * N[(1.0 + N[(0.5 * t$95$2 + N[(0.5 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$3 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$4 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0054000000000000003

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 64.5%

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

   8. Applied rewrites 64.5%

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

   if -0.0054000000000000003 < y < 0.00294999999999999993

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 52.3%

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

   if 0.00294999999999999993 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

   6. Applied rewrites 64.5%

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

Alternative 14: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos x \cdot \left(\sqrt{5} - 1\right)\\ t_3 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_1, t\_2\right), 1\right)\\ t_4 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.0054:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y \cdot t\_0, \cos y - \cos x, -2\right)}{-3 \cdot t\_3}\\ \mathbf{elif}\;y \leq 0.00295:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_4}{\mathsf{fma}\left(-0.75, {y}^{2} \cdot t\_1, 3 \cdot \left(1 + \mathsf{fma}\left(0.5, t\_2, 0.5 \cdot t\_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_4 \cdot t\_0, \sin y, 2\right)}{t\_3}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* (cos x) (- (sqrt 5.0) 1.0)))
        (t_3 (fma 0.5 (fma (cos y) t_1 t_2) 1.0))
        (t_4 (- (cos x) (cos y))))
   (if (<= y -0.0054)
     (/ (fma (* (sin y) t_0) (- (cos y) (cos x)) -2.0) (* -3.0 t_3))
     (if (<= y 0.00295)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_4))
        (fma
         -0.75
         (* (pow y 2.0) t_1)
         (* 3.0 (+ 1.0 (fma 0.5 t_2 (* 0.5 t_1))))))
       (/ (/ (fma (* t_4 t_0) (sin y) 2.0) t_3) 3.0)))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x)) * sqrt(2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = cos(x) * (sqrt(5.0) - 1.0);
	double t_3 = fma(0.5, fma(cos(y), t_1, t_2), 1.0);
	double t_4 = cos(x) - cos(y);
	double tmp;
	if (y <= -0.0054) {
		tmp = fma((sin(y) * t_0), (cos(y) - cos(x)), -2.0) / (-3.0 * t_3);
	} else if (y <= 0.00295) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_4)) / fma(-0.75, (pow(y, 2.0) * t_1), (3.0 * (1.0 + fma(0.5, t_2, (0.5 * t_1)))));
	} else {
		tmp = (fma((t_4 * t_0), sin(y), 2.0) / t_3) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(cos(x) * Float64(sqrt(5.0) - 1.0))
	t_3 = fma(0.5, fma(cos(y), t_1, t_2), 1.0)
	t_4 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (y <= -0.0054)
		tmp = Float64(fma(Float64(sin(y) * t_0), Float64(cos(y) - cos(x)), -2.0) / Float64(-3.0 * t_3));
	elseif (y <= 0.00295)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_4)) / fma(-0.75, Float64((y ^ 2.0) * t_1), Float64(3.0 * Float64(1.0 + fma(0.5, t_2, Float64(0.5 * t_1))))));
	else
		tmp = Float64(Float64(fma(Float64(t_4 * t_0), sin(y), 2.0) / t_3) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0054], N[(N[(N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(-3.0 * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00295], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] / N[(-0.75 * N[(N[Power[y, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(3.0 * N[(1.0 + N[(0.5 * t$95$2 + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$4 * t$95$0), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0054000000000000003

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

   6. Applied rewrites 64.5%

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

   if -0.0054000000000000003 < y < 0.00294999999999999993

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 52.3%

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

   if 0.00294999999999999993 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

   6. Applied rewrites 64.5%

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

Alternative 15: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x \cdot \left(\sqrt{5} - 1\right)\\ t_2 := \frac{\mathsf{fma}\left(\sin y \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), \cos y - \cos x, -2\right)}{-3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, t\_1\right), 1\right)}\\ \mathbf{if}\;y \leq -0.0054:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(-0.75, {y}^{2} \cdot t\_0, 3 \cdot \left(1 + \mathsf{fma}\left(0.5, t\_1, 0.5 \cdot t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (cos x) (- (sqrt 5.0) 1.0)))
        (t_2
         (/
          (fma
           (* (sin y) (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
           (- (cos y) (cos x))
           -2.0)
          (* -3.0 (fma 0.5 (fma (cos y) t_0 t_1) 1.0)))))
   (if (<= y -0.0054)
     t_2
     (if (<= y 1.75e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          (- (cos x) (cos y))))
        (fma
         -0.75
         (* (pow y 2.0) t_0)
         (* 3.0 (+ 1.0 (fma 0.5 t_1 (* 0.5 t_0))))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) * (sqrt(5.0) - 1.0);
	double t_2 = fma((sin(y) * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), (cos(y) - cos(x)), -2.0) / (-3.0 * fma(0.5, fma(cos(y), t_0, t_1), 1.0));
	double tmp;
	if (y <= -0.0054) {
		tmp = t_2;
	} else if (y <= 1.75e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / fma(-0.75, (pow(y, 2.0) * t_0), (3.0 * (1.0 + fma(0.5, t_1, (0.5 * t_0)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) * Float64(sqrt(5.0) - 1.0))
	t_2 = Float64(fma(Float64(sin(y) * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), Float64(cos(y) - cos(x)), -2.0) / Float64(-3.0 * fma(0.5, fma(cos(y), t_0, t_1), 1.0)))
	tmp = 0.0
	if (y <= -0.0054)
		tmp = t_2;
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / fma(-0.75, Float64((y ^ 2.0) * t_0), Float64(3.0 * Float64(1.0 + fma(0.5, t_1, Float64(0.5 * t_0))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(-3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0054], t$95$2, If[LessEqual[y, 1.75e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.75 * N[(N[Power[y, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(3.0 * N[(1.0 + N[(0.5 * t$95$1 + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0054000000000000003 or 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

   6. Applied rewrites 64.5%

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

   if -0.0054000000000000003 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 52.3%

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

Alternative 16: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} - 1\right)\\ t_1 := \cos x - \cos y\\ t_2 := \mathsf{fma}\left(t\_1 \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), \sin y, 2\right)\\ t_3 := 3 - \sqrt{5}\\ t_4 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_3, t\_0\right), 1\right)\\ \mathbf{if}\;y \leq -0.0054:\\ \;\;\;\;t\_2 \cdot \frac{0.3333333333333333}{t\_4}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{\mathsf{fma}\left(-0.75, {y}^{2} \cdot t\_3, 3 \cdot \left(1 + \mathsf{fma}\left(0.5, t\_0, 0.5 \cdot t\_3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 \cdot 0.3333333333333333}{t\_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (- (sqrt 5.0) 1.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (fma
          (* t_1 (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
          (sin y)
          2.0))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (fma 0.5 (fma (cos y) t_3 t_0) 1.0)))
   (if (<= y -0.0054)
     (* t_2 (/ 0.3333333333333333 t_4))
     (if (<= y 1.75e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_1))
        (fma
         -0.75
         (* (pow y 2.0) t_3)
         (* 3.0 (+ 1.0 (fma 0.5 t_0 (* 0.5 t_3))))))
       (/ (* t_2 0.3333333333333333) t_4)))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sqrt(5.0) - 1.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = fma((t_1 * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), sin(y), 2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = fma(0.5, fma(cos(y), t_3, t_0), 1.0);
	double tmp;
	if (y <= -0.0054) {
		tmp = t_2 * (0.3333333333333333 / t_4);
	} else if (y <= 1.75e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_1)) / fma(-0.75, (pow(y, 2.0) * t_3), (3.0 * (1.0 + fma(0.5, t_0, (0.5 * t_3)))));
	} else {
		tmp = (t_2 * 0.3333333333333333) / t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sqrt(5.0) - 1.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = fma(Float64(t_1 * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), sin(y), 2.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = fma(0.5, fma(cos(y), t_3, t_0), 1.0)
	tmp = 0.0
	if (y <= -0.0054)
		tmp = Float64(t_2 * Float64(0.3333333333333333 / t_4));
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_1)) / fma(-0.75, Float64((y ^ 2.0) * t_3), Float64(3.0 * Float64(1.0 + fma(0.5, t_0, Float64(0.5 * t_3))))));
	else
		tmp = Float64(Float64(t_2 * 0.3333333333333333) / t_4);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$3 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -0.0054], N[(t$95$2 * N[(0.3333333333333333 / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(-0.75 * N[(N[Power[y, 2.0], $MachinePrecision] * t$95$3), $MachinePrecision] + N[(3.0 * N[(1.0 + N[(0.5 * t$95$0 + N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * 0.3333333333333333), $MachinePrecision] / t$95$4), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0054000000000000003

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

   6. Applied rewrites 64.4%

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

   if -0.0054000000000000003 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 52.3%

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

   if 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

   6. Applied rewrites 64.4%

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

Alternative 17: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} - 1\right)\\ t_1 := \cos x - \cos y\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(t\_1 \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), \sin y, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_2, t\_0\right), 1\right)}\\ \mathbf{if}\;y \leq -0.0054:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{\mathsf{fma}\left(-0.75, {y}^{2} \cdot t\_2, 3 \cdot \left(1 + \mathsf{fma}\left(0.5, t\_0, 0.5 \cdot t\_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (- (sqrt 5.0) 1.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (*
          (fma
           (* t_1 (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0)))
           (sin y)
           2.0)
          (/ 0.3333333333333333 (fma 0.5 (fma (cos y) t_2 t_0) 1.0)))))
   (if (<= y -0.0054)
     t_3
     (if (<= y 1.75e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_1))
        (fma
         -0.75
         (* (pow y 2.0) t_2)
         (* 3.0 (+ 1.0 (fma 0.5 t_0 (* 0.5 t_2))))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sqrt(5.0) - 1.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma((t_1 * (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), sin(y), 2.0) * (0.3333333333333333 / fma(0.5, fma(cos(y), t_2, t_0), 1.0));
	double tmp;
	if (y <= -0.0054) {
		tmp = t_3;
	} else if (y <= 1.75e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_1)) / fma(-0.75, (pow(y, 2.0) * t_2), (3.0 * (1.0 + fma(0.5, t_0, (0.5 * t_2)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sqrt(5.0) - 1.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(fma(Float64(t_1 * Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0))), sin(y), 2.0) * Float64(0.3333333333333333 / fma(0.5, fma(cos(y), t_2, t_0), 1.0)))
	tmp = 0.0
	if (y <= -0.0054)
		tmp = t_3;
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_1)) / fma(-0.75, Float64((y ^ 2.0) * t_2), Float64(3.0 * Float64(1.0 + fma(0.5, t_0, Float64(0.5 * t_2))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0054], t$95$3, If[LessEqual[y, 1.75e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(-0.75 * N[(N[Power[y, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(3.0 * N[(1.0 + N[(0.5 * t$95$0 + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0054000000000000003 or 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

   6. Applied rewrites 64.4%

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

   if -0.0054000000000000003 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 52.3%

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

Alternative 18: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.0054:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(-0.75, {y}^{2} \cdot t\_0, 3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, 0.5 \cdot t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (+
           2.0
           (*
            (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y))
            (- 1.0 (cos y))))
          (*
           3.0
           (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))))
   (if (<= y -0.0054)
     t_2
     (if (<= y 1.75e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          (- (cos x) (cos y))))
        (fma
         -0.75
         (* (pow y 2.0) t_0)
         (* 3.0 (+ 1.0 (fma 0.5 (* (cos x) t_1) (* 0.5 t_0))))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * (1.0 - cos(y)))) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	double tmp;
	if (y <= -0.0054) {
		tmp = t_2;
	} else if (y <= 1.75e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / fma(-0.75, (pow(y, 2.0) * t_0), (3.0 * (1.0 + fma(0.5, (cos(x) * t_1), (0.5 * t_0)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -0.0054)
		tmp = t_2;
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / fma(-0.75, Float64((y ^ 2.0) * t_0), Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), Float64(0.5 * t_0))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0054], t$95$2, If[LessEqual[y, 1.75e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.75 * N[(N[Power[y, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0054000000000000003 or 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 62.9

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

   7. Applied rewrites 62.9%

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

   if -0.0054000000000000003 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 52.3%

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

Alternative 19: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.0055:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 1, t\_1 \cdot \cos y\right), 0.5, 1\right)}}{3}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_2}{\mathsf{fma}\left(-0.75, {y}^{2} \cdot t\_1, 3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_0, 0.5 \cdot t\_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot t\_2}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{t\_1}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (cos x) (cos y))))
   (if (<= y -0.0055)
     (/
      (/
       (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 (fma t_0 1.0 (* t_1 (cos y))) 0.5 1.0))
      3.0)
     (if (<= y 1.75e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_2))
        (fma
         -0.75
         (* (pow y 2.0) t_1)
         (* 3.0 (+ 1.0 (fma 0.5 (* (cos x) t_0) (* 0.5 t_1))))))
       (/
        (+ 2.0 (* (* -0.0625 (* (pow (sin y) 2.0) (sqrt 2.0))) t_2))
        (*
         3.0
         (+ (+ 1.0 (* (/ t_0 2.0) (cos x))) (* (/ t_1 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = cos(x) - cos(y);
	double tmp;
	if (y <= -0.0055) {
		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(fma(t_0, 1.0, (t_1 * cos(y))), 0.5, 1.0)) / 3.0;
	} else if (y <= 1.75e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_2)) / fma(-0.75, (pow(y, 2.0) * t_1), (3.0 * (1.0 + fma(0.5, (cos(x) * t_0), (0.5 * t_1)))));
	} else {
		tmp = (2.0 + ((-0.0625 * (pow(sin(y), 2.0) * sqrt(2.0))) * t_2)) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + ((t_1 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (y <= -0.0055)
		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) / fma(fma(t_0, 1.0, Float64(t_1 * cos(y))), 0.5, 1.0)) / 3.0);
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_2)) / fma(-0.75, Float64((y ^ 2.0) * t_1), Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(x) * t_0), Float64(0.5 * t_1))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * sqrt(2.0))) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(t_1 / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0055], 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[(t$95$0 * 1.0 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[y, 1.75e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(-0.75 * N[(N[Power[y, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * 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$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0054999999999999997

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.4%

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

   if -0.0054999999999999997 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 52.3%

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

   if 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.8

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

   4. Applied rewrites 62.8%

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

Alternative 20: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.0055:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(1 - \cos y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos y, 1 \cdot t\_2\right), 0.5, 1\right)}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{\mathsf{fma}\left(-0.75, {y}^{2} \cdot t\_0, 3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, 0.5 \cdot t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= y -0.0055)
     (*
      0.3333333333333333
      (/
       (fma
        (* (fma -0.0625 (sin x) (sin y)) (- 1.0 (cos y)))
        (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0))
        2.0)
       (fma (fma t_0 (cos y) (* 1.0 t_2)) 0.5 1.0)))
     (if (<= y 1.75e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_1))
        (fma
         -0.75
         (* (pow y 2.0) t_0)
         (* 3.0 (+ 1.0 (fma 0.5 (* (cos x) t_2) (* 0.5 t_0))))))
       (/
        (+ 2.0 (* (* -0.0625 (* (pow (sin y) 2.0) (sqrt 2.0))) t_1))
        (*
         3.0
         (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.0055) {
		tmp = 0.3333333333333333 * (fma((fma(-0.0625, sin(x), sin(y)) * (1.0 - cos(y))), (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), 2.0) / fma(fma(t_0, cos(y), (1.0 * t_2)), 0.5, 1.0));
	} else if (y <= 1.75e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_1)) / fma(-0.75, (pow(y, 2.0) * t_0), (3.0 * (1.0 + fma(0.5, (cos(x) * t_2), (0.5 * t_0)))));
	} else {
		tmp = (2.0 + ((-0.0625 * (pow(sin(y), 2.0) * sqrt(2.0))) * t_1)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.0055)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(fma(-0.0625, sin(x), sin(y)) * Float64(1.0 - cos(y))), Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), 2.0) / fma(fma(t_0, cos(y), Float64(1.0 * t_2)), 0.5, 1.0)));
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_1)) / fma(-0.75, Float64((y ^ 2.0) * t_0), Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), Float64(0.5 * t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * sqrt(2.0))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.0055], N[(0.3333333333333333 * N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(1.0 * t$95$2), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(-0.75 * N[(N[Power[y, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0054999999999999997

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.3%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

   13. Applied rewrites 60.3%

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

   if -0.0054999999999999997 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 52.3%

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

   if 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.8

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

   4. Applied rewrites 62.8%

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

Alternative 21: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(1 - \cos y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos y, 1 \cdot t\_2\right), 0.5, 1\right)}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, 0.5 \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= y -6.5e-8)
     (*
      0.3333333333333333
      (/
       (fma
        (* (fma -0.0625 (sin x) (sin y)) (- 1.0 (cos y)))
        (* (fma (sin y) -0.0625 (sin x)) (sqrt 2.0))
        2.0)
       (fma (fma t_0 (cos y) (* 1.0 t_2)) 0.5 1.0)))
     (if (<= y 1.75e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_1))
        (* 3.0 (+ 1.0 (fma 0.5 (* (cos x) t_2) (* 0.5 t_0)))))
       (/
        (+ 2.0 (* (* -0.0625 (* (pow (sin y) 2.0) (sqrt 2.0))) t_1))
        (*
         3.0
         (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -6.5e-8) {
		tmp = 0.3333333333333333 * (fma((fma(-0.0625, sin(x), sin(y)) * (1.0 - cos(y))), (fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), 2.0) / fma(fma(t_0, cos(y), (1.0 * t_2)), 0.5, 1.0));
	} else if (y <= 1.75e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_1)) / (3.0 * (1.0 + fma(0.5, (cos(x) * t_2), (0.5 * t_0))));
	} else {
		tmp = (2.0 + ((-0.0625 * (pow(sin(y), 2.0) * sqrt(2.0))) * t_1)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -6.5e-8)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(fma(-0.0625, sin(x), sin(y)) * Float64(1.0 - cos(y))), Float64(fma(sin(y), -0.0625, sin(x)) * sqrt(2.0)), 2.0) / fma(fma(t_0, cos(y), Float64(1.0 * t_2)), 0.5, 1.0)));
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), Float64(0.5 * t_0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * sqrt(2.0))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -6.5e-8], N[(0.3333333333333333 * N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(1.0 * t$95$2), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.49999999999999997e-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%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.3%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

   13. Applied rewrites 60.3%

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

   if -6.49999999999999997e-8 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 49.6

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

   10. Applied rewrites 49.6%

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

   if 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.8

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

   4. Applied rewrites 62.8%

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

Alternative 22: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := 3 - \sqrt{5}\\ t_3 := \frac{2 + \left(-0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{t\_2}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_0, 0.5 \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (/
          (+ 2.0 (* (* -0.0625 (* (pow (sin y) 2.0) (sqrt 2.0))) t_1))
          (*
           3.0
           (+ (+ 1.0 (* (/ t_0 2.0) (cos x))) (* (/ t_2 2.0) (cos y)))))))
   (if (<= y -6.5e-8)
     t_3
     (if (<= y 1.75e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_1))
        (* 3.0 (+ 1.0 (fma 0.5 (* (cos x) t_0) (* 0.5 t_2)))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = (2.0 + ((-0.0625 * (pow(sin(y), 2.0) * sqrt(2.0))) * t_1)) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + ((t_2 / 2.0) * cos(y))));
	double tmp;
	if (y <= -6.5e-8) {
		tmp = t_3;
	} else if (y <= 1.75e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_1)) / (3.0 * (1.0 + fma(0.5, (cos(x) * t_0), (0.5 * t_2))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * sqrt(2.0))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(t_2 / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -6.5e-8)
		tmp = t_3;
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(x) * t_0), Float64(0.5 * t_2)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e-8], t$95$3, If[LessEqual[y, 1.75e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.49999999999999997e-8 or 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 62.8

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

   4. Applied rewrites 62.8%

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

   if -6.49999999999999997e-8 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 49.6

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

   10. Applied rewrites 49.6%

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

Alternative 23: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := {\sin y}^{2}\\ t_3 := \cos x - \cos y\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_3}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, 0.5 \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_3 \cdot \sqrt{2}, -0.0625 \cdot t\_2, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_0 \cdot \cos y\right), 0.5, 1\right)}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (pow (sin y) 2.0))
        (t_3 (- (cos x) (cos y))))
   (if (<= y -6.5e-8)
     (/
      (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) (- 1.0 (cos y))))))
      (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))
     (if (<= y 1.75e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_3))
        (* 3.0 (+ 1.0 (fma 0.5 (* (cos x) t_1) (* 0.5 t_0)))))
       (/
        (/
         (fma (* t_3 (sqrt 2.0)) (* -0.0625 t_2) 2.0)
         (fma (fma t_1 (cos x) (* t_0 (cos y))) 0.5 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = pow(sin(y), 2.0);
	double t_3 = cos(x) - cos(y);
	double tmp;
	if (y <= -6.5e-8) {
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	} else if (y <= 1.75e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_3)) / (3.0 * (1.0 + fma(0.5, (cos(x) * t_1), (0.5 * t_0))));
	} else {
		tmp = (fma((t_3 * sqrt(2.0)), (-0.0625 * t_2), 2.0) / fma(fma(t_1, cos(x), (t_0 * cos(y))), 0.5, 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = sin(y) ^ 2.0
	t_3 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (y <= -6.5e-8)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * t_3)) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), Float64(0.5 * t_0)))));
	else
		tmp = Float64(Float64(fma(Float64(t_3 * sqrt(2.0)), Float64(-0.0625 * t_2), 2.0) / fma(fma(t_1, cos(x), Float64(t_0 * cos(y))), 0.5, 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e-8], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.49999999999999997e-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 x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 62.8

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

   4. Applied rewrites 62.8%

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

   if -6.49999999999999997e-8 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 49.6

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

   10. Applied rewrites 49.6%

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

   if 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.8

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

   6. Applied rewrites 62.8%

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

Alternative 24: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\\ t_2 := \sqrt{5} - 1\\ t_3 := 3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + t\_1}{t\_3}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, 0.5 \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + t\_1\right) + 1}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3
         (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) (* (/ t_0 2.0) (cos y))))))
   (if (<= y -6.5e-8)
     (/ (+ 2.0 t_1) t_3)
     (if (<= y 1.75e-7)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          (- (cos x) (cos y))))
        (* 3.0 (+ 1.0 (fma 0.5 (* (cos x) t_2) (* 0.5 t_0)))))
       (/ (+ (+ 1.0 t_1) 1.0) t_3)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = -0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y))));
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y)));
	double tmp;
	if (y <= -6.5e-8) {
		tmp = (2.0 + t_1) / t_3;
	} else if (y <= 1.75e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * (1.0 + fma(0.5, (cos(x) * t_2), (0.5 * t_0))));
	} else {
		tmp = ((1.0 + t_1) + 1.0) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = 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 = Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y))))
	tmp = 0.0
	if (y <= -6.5e-8)
		tmp = Float64(Float64(2.0 + t_1) / t_3);
	elseif (y <= 1.75e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), Float64(0.5 * t_0)))));
	else
		tmp = Float64(Float64(Float64(1.0 + t_1) + 1.0) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[(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]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e-8], N[(N[(2.0 + t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 1.75e-7], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.49999999999999997e-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 x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 62.8

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

   4. Applied rewrites 62.8%

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

   if -6.49999999999999997e-8 < y < 1.74999999999999992e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 49.6

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

   10. Applied rewrites 49.6%

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

   if 1.74999999999999992e-7 < y

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 64.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 62.8

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

   9. Applied rewrites 62.8%

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

Alternative 25: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_1\right), 1\right)}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot 1\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (*
           0.3333333333333333
           (+
            2.0
            (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
          (fma 0.5 (fma (cos y) t_0 (* (cos x) t_1)) 1.0))))
   (if (<= x -2.4e-5)
     t_2
     (if (<= x 5.6e-5)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y))
          (- 1.0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) 1.0)) (* (/ t_0 2.0) (cos y)))))
       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 = (0.3333333333333333 * (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0)))))) / fma(0.5, fma(cos(y), t_0, (cos(x) * t_1)), 1.0);
	double tmp;
	if (x <= -2.4e-5) {
		tmp = t_2;
	} else if (x <= 5.6e-5) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * (1.0 - cos(y)))) / (3.0 * ((1.0 + ((t_1 / 2.0) * 1.0)) + ((t_0 / 2.0) * cos(y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))) / fma(0.5, fma(cos(y), t_0, Float64(cos(x) * t_1)), 1.0))
	tmp = 0.0
	if (x <= -2.4e-5)
		tmp = t_2;
	elseif (x <= 5.6e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * 1.0)) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.3333333333333333 * N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4e-5], t$95$2, If[LessEqual[x, 5.6e-5], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000001e-5 or 5.59999999999999992e-5 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 62.0

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

   8. Applied rewrites 62.0%

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

   if -2.4000000000000001e-5 < x < 5.59999999999999992e-5

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.5

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 62.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.3%

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

Alternative 26: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_1\right), 1\right)}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.00047:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(x + -0.0625 \cdot \sin y\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, 1 \cdot t\_1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (*
           0.3333333333333333
           (+
            2.0
            (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
          (fma 0.5 (fma (cos y) t_0 (* (cos x) t_1)) 1.0))))
   (if (<= x -2.6e-5)
     t_2
     (if (<= x 0.00047)
       (/
        (*
         (fma
          (* (* (- 1.0 (cos y)) (sqrt 2.0)) (+ x (* -0.0625 (sin y))))
          (fma -0.0625 (sin x) (sin y))
          2.0)
         0.3333333333333333)
        (fma 0.5 (fma (cos y) t_0 (* 1.0 t_1)) 1.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (0.3333333333333333 * (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0)))))) / fma(0.5, fma(cos(y), t_0, (cos(x) * t_1)), 1.0);
	double tmp;
	if (x <= -2.6e-5) {
		tmp = t_2;
	} else if (x <= 0.00047) {
		tmp = (fma((((1.0 - cos(y)) * sqrt(2.0)) * (x + (-0.0625 * sin(y)))), fma(-0.0625, sin(x), sin(y)), 2.0) * 0.3333333333333333) / fma(0.5, fma(cos(y), t_0, (1.0 * t_1)), 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))) / fma(0.5, fma(cos(y), t_0, Float64(cos(x) * t_1)), 1.0))
	tmp = 0.0
	if (x <= -2.6e-5)
		tmp = t_2;
	elseif (x <= 0.00047)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * Float64(x + Float64(-0.0625 * sin(y)))), fma(-0.0625, sin(x), sin(y)), 2.0) * 0.3333333333333333) / fma(0.5, fma(cos(y), t_0, Float64(1.0 * t_1)), 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.3333333333333333 * N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.6e-5], t$95$2, If[LessEqual[x, 0.00047], N[(N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(1.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.59999999999999984e-5 or 4.69999999999999986e-4 < 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%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 62.0

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

   8. Applied rewrites 62.0%

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

   if -2.59999999999999984e-5 < x < 4.69999999999999986e-4

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.3%

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

   12. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 55.8

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

   14. Applied rewrites 55.8%

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

Alternative 27: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + 0.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_0\right) - 1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (*
           0.3333333333333333
           (+
            2.0
            (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
          (fma 0.5 (fma (cos y) t_0 (* (cos x) (- (sqrt 5.0) 1.0))) 1.0))))
   (if (<= x -2.6e-6)
     t_1
     (if (<= x 1.45e-5)
       (/
        (/
         (+
          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_0)) 1.0))))
        3.0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (0.3333333333333333 * (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0)))))) / fma(0.5, fma(cos(y), t_0, (cos(x) * (sqrt(5.0) - 1.0))), 1.0);
	double tmp;
	if (x <= -2.6e-6) {
		tmp = t_1;
	} else if (x <= 1.45e-5) {
		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_0)) - 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(0.3333333333333333 * Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))) / fma(0.5, fma(cos(y), t_0, Float64(cos(x) * Float64(sqrt(5.0) - 1.0))), 1.0))
	tmp = 0.0
	if (x <= -2.6e-6)
		tmp = t_1;
	elseif (x <= 1.45e-5)
		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_0)) - 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[(0.3333333333333333 * N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.6e-6], t$95$1, If[LessEqual[x, 1.45e-5], 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$0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000009e-6 or 1.45e-5 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 62.0

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

   8. Applied rewrites 62.0%

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

   if -2.60000000000000009e-6 < x < 1.45e-5

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   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 59.9%

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

Alternative 28: 78.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(x + x\right), -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_0\right), -1\right)}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7800:\\ \;\;\;\;\frac{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + 0.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_0\right) - 1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (+ x x))))
            -2.0)
           0.3333333333333333)
          (fma -0.5 (fma (cos x) (- (sqrt 5.0) 1.0) t_0) -1.0))))
   (if (<= x -1.25e-5)
     t_1
     (if (<= x 7800.0)
       (/
        (/
         (+
          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_0)) 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((x + x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma(cos(x), (sqrt(5.0) - 1.0), t_0), -1.0);
	double tmp;
	if (x <= -1.25e-5) {
		tmp = t_1;
	} else if (x <= 7800.0) {
		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_0)) - 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(x + x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), t_0), -1.0))
	tmp = 0.0
	if (x <= -1.25e-5)
		tmp = t_1;
	elseif (x <= 7800.0)
		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_0)) - 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[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e-5], t$95$1, If[LessEqual[x, 7800.0], 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$0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25000000000000006e-5 or 7800 < 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%

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

   6. Applied rewrites 59.6%

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

   if -1.25000000000000006e-5 < x < 7800

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   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 59.9%

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

Alternative 29: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(x + x\right), -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_0\right), -1\right)}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7800:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_0, 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (*
           (fma
            (* 0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
            (- 0.5 (* 0.5 (cos (+ x x))))
            -2.0)
           0.3333333333333333)
          (fma -0.5 (fma (cos x) (- (sqrt 5.0) 1.0) t_0) -1.0))))
   (if (<= x -1.25e-5)
     t_1
     (if (<= x 7800.0)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_0) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0))))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (fma((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((x + x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma(cos(x), (sqrt(5.0) - 1.0), t_0), -1.0);
	double tmp;
	if (x <= -1.25e-5) {
		tmp = t_1;
	} else if (x <= 7800.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_0), (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(fma(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), t_0), -1.0))
	tmp = 0.0
	if (x <= -1.25e-5)
		tmp = t_1;
	elseif (x <= 7800.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_0), Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(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[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e-5], t$95$1, If[LessEqual[x, 7800.0], 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[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25000000000000006e-5 or 7800 < 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%

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

   6. Applied rewrites 59.6%

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

   if -1.25000000000000006e-5 < x < 7800

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 59.9%

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

Alternative 30: 59.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

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(x + x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), -1.0))
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[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

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%

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

6. Applied rewrites 59.6%

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

Alternative 31: 59.6% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
    (- 0.5 (* 0.5 (cos (+ x x))))
    2.0)
   (fma (fma (cos x) (- (sqrt 5.0) 1.0) (- 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((x + x)))), 2.0) / fma(fma(cos(x), (sqrt(5.0) - 1.0), (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(x + x)))), 2.0) / fma(fma(cos(x), Float64(sqrt(5.0) - 1.0), 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[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $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%

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

6. Applied rewrites 59.6%

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

Alternative 32: 59.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(x + x\right), 2\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 2.5\right)\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
    (- 0.5 (* 0.5 (cos (+ x x))))
    2.0)
   (fma -0.5 (sqrt 5.0) (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 2.5)))
  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((x + x)))), 2.0) / fma(-0.5, sqrt(5.0), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 2.5))) * 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(x + x)))), 2.0) / fma(-0.5, sqrt(5.0), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 2.5))) * 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[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 2.5), $MachinePrecision]), $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

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

   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%

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

8. Applied rewrites 59.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(x + x\right), 2\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 2.5\right)\right)} \cdot \color{blue}{0.3333333333333333} \]
9. Add Preprocessing

Alternative 33: 43.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right), 0.5, 2.5\right) - 0.5 \cdot \sqrt{5}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/ 2.0 (- (fma (* (cos x) (- (sqrt 5.0) 1.0)) 0.5 2.5) (* 0.5 (sqrt 5.0))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * (2.0 / (fma((cos(x) * (sqrt(5.0) - 1.0)), 0.5, 2.5) - (0.5 * sqrt(5.0))));
}

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(2.0 / Float64(fma(Float64(cos(x) * Float64(sqrt(5.0) - 1.0)), 0.5, 2.5) - Float64(0.5 * sqrt(5.0)))))
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(2.0 / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 2.5), $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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. lift-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-in N/A

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

   9. sub-flip N/A

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

   10. lift--.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval 59.6

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right), 0.5, 2.5\right) - 0.5 \cdot \sqrt{5}} \]

8. Applied rewrites 59.6%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right), 0.5, 2.5\right) - \color{blue}{0.5} \cdot \sqrt{5}} \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 43.1%

    \[\leadsto 0.3333333333333333 \cdot \frac{2}{\color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right), 0.5, 2.5\right)} - 0.5 \cdot \sqrt{5}} \]
12. Add Preprocessing

Alternative 34: 40.7% 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 40.7%

   \[\leadsto 0.3333333333333333 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(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))))))