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

Percentage Accurate: 99.3% → 99.4%

Time: 9.1s

Alternatives: 25

Speedup: 1.1×

Specification

Math

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

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

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

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, 1.1× speedup

Math

\[\frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(1.2360679774997898, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (* (- (cos x) (cos y)) (sqrt 2.0)) (fma (sin y) -0.0625 (sin x)))
   (fma -0.0625 (sin x) (sin y))
   2.0)
  (fma
   (fma 1.2360679774997898 (cos x) (* 0.7639320225002103 (cos y)))
   1.5
   3.0)))

C

double code(double x, double y) {
	return fma((((cos(x) - cos(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(fma(1.2360679774997898, cos(x), (0.7639320225002103 * cos(y))), 1.5, 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(fma(1.2360679774997898, cos(x), Float64(0.7639320225002103 * cos(y))), 1.5, 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.2360679774997898 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

6. Evaluated real constant 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

7. Evaluated real constant 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{1.2360679774997898}, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
8. Add Preprocessing

Alternative 2: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(\cos x - \cos y\right) \cdot \sqrt{2}\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -220000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2, \sin y \cdot t\_0, 2\right)}{\frac{\mathsf{fma}\left(t\_1, \cos x, t\_3 \cdot \cos y\right)}{2} + 1}}{3}\\ \mathbf{elif}\;y \leq 0.024:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos x, \left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \sqrt{2}\right) \cdot t\_0, -2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_3 \cdot \sin \left(\left(-y\right) + \pi \cdot 0.5\right)\right), 1\right)} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot t\_0, \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)}\\ \end{array} \]

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)) (sqrt 2.0)))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= y -220000.0)
     (/
      (/
       (fma t_2 (* (sin y) t_0) 2.0)
       (+ (/ (fma t_1 (cos x) (* t_3 (cos y))) 2.0) 1.0))
      3.0)
     (if (<= y 0.024)
       (*
        (/
         (fma
          (- 1.0 (cos x))
          (* (* (fma -0.0625 (sin x) (sin y)) (sqrt 2.0)) t_0)
          -2.0)
         (fma 0.5 (fma t_1 (cos x) (* t_3 (sin (+ (- y) (* PI 0.5))))) 1.0))
        -0.3333333333333333)
       (/
        (fma (* t_2 t_0) (sin y) 2.0)
        (fma (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 1.5 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)) * sqrt(2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -220000.0) {
		tmp = (fma(t_2, (sin(y) * t_0), 2.0) / ((fma(t_1, cos(x), (t_3 * cos(y))) / 2.0) + 1.0)) / 3.0;
	} else if (y <= 0.024) {
		tmp = (fma((1.0 - cos(x)), ((fma(-0.0625, sin(x), sin(y)) * sqrt(2.0)) * t_0), -2.0) / fma(0.5, fma(t_1, cos(x), (t_3 * sin((-y + (((double) M_PI) * 0.5))))), 1.0)) * -0.3333333333333333;
	} else {
		tmp = fma((t_2 * t_0), sin(y), 2.0) / fma(fma(t_1, cos(x), (0.7639320225002103 * cos(y))), 1.5, 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(Float64(cos(x) - cos(y)) * sqrt(2.0))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -220000.0)
		tmp = Float64(Float64(fma(t_2, Float64(sin(y) * t_0), 2.0) / Float64(Float64(fma(t_1, cos(x), Float64(t_3 * cos(y))) / 2.0) + 1.0)) / 3.0);
	elseif (y <= 0.024)
		tmp = Float64(Float64(fma(Float64(1.0 - cos(x)), Float64(Float64(fma(-0.0625, sin(x), sin(y)) * sqrt(2.0)) * t_0), -2.0) / fma(0.5, fma(t_1, cos(x), Float64(t_3 * sin(Float64(Float64(-y) + Float64(pi * 0.5))))), 1.0)) * -0.3333333333333333);
	else
		tmp = Float64(fma(Float64(t_2 * t_0), sin(y), 2.0) / fma(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))), 1.5, 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[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -220000.0], N[(N[(N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[y, 0.024], N[(N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + -2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$3 * N[Sin[N[((-y) + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(N[(t$95$2 * t$95$0), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e5

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.2%

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

   6. Applied rewrites 64.2%

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

   if -2.2e5 < y < 0.024

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 62.5%

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

   8. Applied rewrites 62.5%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 59.8%

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

   10. Applied rewrites 59.8%

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

   if 0.024 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-sin.f64 64.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 64.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(\cos x - \cos y\right) \cdot \sqrt{2}\\ t_3 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ \mathbf{if}\;y \leq -220000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2, \sin y \cdot t\_0, 2\right)}{\frac{\mathsf{fma}\left(t\_1, \cos x, t\_3\right)}{2} + 1}}{3}\\ \mathbf{elif}\;y \leq 0.024:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos x, \left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \sqrt{2}\right) \cdot t\_0, -2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(1.2360679774997898, \cos x, t\_3\right), 1\right)} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot t\_0, \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)}\\ \end{array} \]

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)) (sqrt 2.0)))
        (t_3 (* (- 3.0 (sqrt 5.0)) (cos y))))
   (if (<= y -220000.0)
     (/
      (/ (fma t_2 (* (sin y) t_0) 2.0) (+ (/ (fma t_1 (cos x) t_3) 2.0) 1.0))
      3.0)
     (if (<= y 0.024)
       (*
        (/
         (fma
          (- 1.0 (cos x))
          (* (* (fma -0.0625 (sin x) (sin y)) (sqrt 2.0)) t_0)
          -2.0)
         (fma 0.5 (fma 1.2360679774997898 (cos x) t_3) 1.0))
        -0.3333333333333333)
       (/
        (fma (* t_2 t_0) (sin y) 2.0)
        (fma (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 1.5 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)) * sqrt(2.0);
	double t_3 = (3.0 - sqrt(5.0)) * cos(y);
	double tmp;
	if (y <= -220000.0) {
		tmp = (fma(t_2, (sin(y) * t_0), 2.0) / ((fma(t_1, cos(x), t_3) / 2.0) + 1.0)) / 3.0;
	} else if (y <= 0.024) {
		tmp = (fma((1.0 - cos(x)), ((fma(-0.0625, sin(x), sin(y)) * sqrt(2.0)) * t_0), -2.0) / fma(0.5, fma(1.2360679774997898, cos(x), t_3), 1.0)) * -0.3333333333333333;
	} else {
		tmp = fma((t_2 * t_0), sin(y), 2.0) / fma(fma(t_1, cos(x), (0.7639320225002103 * cos(y))), 1.5, 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(Float64(cos(x) - cos(y)) * sqrt(2.0))
	t_3 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	tmp = 0.0
	if (y <= -220000.0)
		tmp = Float64(Float64(fma(t_2, Float64(sin(y) * t_0), 2.0) / Float64(Float64(fma(t_1, cos(x), t_3) / 2.0) + 1.0)) / 3.0);
	elseif (y <= 0.024)
		tmp = Float64(Float64(fma(Float64(1.0 - cos(x)), Float64(Float64(fma(-0.0625, sin(x), sin(y)) * sqrt(2.0)) * t_0), -2.0) / fma(0.5, fma(1.2360679774997898, cos(x), t_3), 1.0)) * -0.3333333333333333);
	else
		tmp = Float64(fma(Float64(t_2 * t_0), sin(y), 2.0) / fma(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))), 1.5, 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[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -220000.0], N[(N[(N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[y, 0.024], N[(N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + -2.0), $MachinePrecision] / N[(0.5 * N[(1.2360679774997898 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(N[(t$95$2 * t$95$0), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e5

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.2%

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

   6. Applied rewrites 64.2%

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

   if -2.2e5 < y < 0.024

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 62.5%

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

   8. Applied rewrites 62.5%

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

   9. Evaluated real constant 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos x, \left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), -2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\color{blue}{1.2360679774997898}, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot -0.3333333333333333 \]

   if 0.024 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-sin.f64 64.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 64.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot t\_0, \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)}\\ \mathbf{if}\;y \leq -220000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.024:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos x, \left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \sqrt{2}\right) \cdot t\_0, -2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(1.2360679774997898, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1
         (/
          (fma (* (* (- (cos x) (cos y)) (sqrt 2.0)) t_0) (sin y) 2.0)
          (fma
           (fma (- (sqrt 5.0) 1.0) (cos x) (* 0.7639320225002103 (cos y)))
           1.5
           3.0))))
   (if (<= y -220000.0)
     t_1
     (if (<= y 0.024)
       (*
        (/
         (fma
          (- 1.0 (cos x))
          (* (* (fma -0.0625 (sin x) (sin y)) (sqrt 2.0)) t_0)
          -2.0)
         (fma
          0.5
          (fma 1.2360679774997898 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          1.0))
        -0.3333333333333333)
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = fma((((cos(x) - cos(y)) * sqrt(2.0)) * t_0), sin(y), 2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), (0.7639320225002103 * cos(y))), 1.5, 3.0);
	double tmp;
	if (y <= -220000.0) {
		tmp = t_1;
	} else if (y <= 0.024) {
		tmp = (fma((1.0 - cos(x)), ((fma(-0.0625, sin(x), sin(y)) * sqrt(2.0)) * t_0), -2.0) / fma(0.5, fma(1.2360679774997898, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0)) * -0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(fma(Float64(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)) * t_0), sin(y), 2.0) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(0.7639320225002103 * cos(y))), 1.5, 3.0))
	tmp = 0.0
	if (y <= -220000.0)
		tmp = t_1;
	elseif (y <= 0.024)
		tmp = Float64(Float64(fma(Float64(1.0 - cos(x)), Float64(Float64(fma(-0.0625, sin(x), sin(y)) * sqrt(2.0)) * t_0), -2.0) / fma(0.5, fma(1.2360679774997898, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)) * -0.3333333333333333);
	else
		tmp = t_1;
	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[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -220000.0], t$95$1, If[LessEqual[y, 0.024], N[(N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + -2.0), $MachinePrecision] / N[(0.5 * N[(1.2360679774997898 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2e5 or 0.024 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-sin.f64 64.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 64.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if -2.2e5 < y < 0.024

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 62.5%

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

   8. Applied rewrites 62.5%

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

   9. Evaluated real constant 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos x, \left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), -2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\color{blue}{1.2360679774997898}, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot -0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\left(t\_0 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)}\\ \mathbf{if}\;y \leq -0.026:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.024:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (fma
           (* (* t_0 (sqrt 2.0)) (fma (sin y) -0.0625 (sin x)))
           (sin y)
           2.0)
          (fma (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 1.5 3.0))))
   (if (<= y -0.026)
     t_2
     (if (<= y 0.024)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          t_0))
        (*
         3.0
         (+
          (+ 1.0 (* (/ t_1 2.0) (cos x)))
          (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma(((t_0 * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), sin(y), 2.0) / fma(fma(t_1, cos(x), (0.7639320225002103 * cos(y))), 1.5, 3.0);
	double tmp;
	if (y <= -0.026) {
		tmp = t_2;
	} else if (y <= 0.024) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * t_0)) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(fma(Float64(Float64(t_0 * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), sin(y), 2.0) / fma(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))), 1.5, 3.0))
	tmp = 0.0
	if (y <= -0.026)
		tmp = t_2;
	elseif (y <= 0.024)
		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_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.026], t$95$2, If[LessEqual[y, 0.024], 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$0), $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[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.025999999999999999 or 0.024 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-sin.f64 64.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sin y, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 64.3%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \color{blue}{\sin y}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if -0.025999999999999999 < y < 0.024

   1. Initial program 99.3%

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

   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 51.1%

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

   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 50.2%

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

Alternative 6: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.024:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot t\_2\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)}\\ \mathbf{elif}\;y \leq 0.068:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\frac{\mathsf{fma}\left(t\_1, 1, t\_0 \cdot \cos y\right)}{2} + 1}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 1.0 (cos y))))
   (if (<= y -0.024)
     (/
      (fma
       (* -0.0625 (* (sin y) (* (sqrt 2.0) t_2)))
       (fma -0.0625 (sin x) (sin y))
       2.0)
      (fma (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 1.5 3.0))
     (if (<= y 0.068)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          (- (cos x) (cos y))))
        (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))
       (/
        (/
         (fma
          (* t_2 (sqrt 2.0))
          (* (fma (sin x) -0.0625 (sin y)) (fma (sin y) -0.0625 (sin x)))
          2.0)
         (+ (/ (fma t_1 1.0 (* t_0 (cos y))) 2.0) 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 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.024) {
		tmp = fma((-0.0625 * (sin(y) * (sqrt(2.0) * t_2))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(fma(t_1, cos(x), (0.7639320225002103 * cos(y))), 1.5, 3.0);
	} else if (y <= 0.068) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	} else {
		tmp = (fma((t_2 * sqrt(2.0)), (fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / ((fma(t_1, 1.0, (t_0 * cos(y))) / 2.0) + 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 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.024)
		tmp = Float64(fma(Float64(-0.0625 * Float64(sin(y) * Float64(sqrt(2.0) * t_2))), fma(-0.0625, sin(x), sin(y)), 2.0) / fma(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))), 1.5, 3.0));
	elseif (y <= 0.068)
		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(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	else
		tmp = Float64(Float64(fma(Float64(t_2 * sqrt(2.0)), Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / Float64(Float64(fma(t_1, 1.0, Float64(t_0 * cos(y))) / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.024], N[(N[(N[(-0.0625 * N[(N[Sin[y], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.068], 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[(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], N[(N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * 1.0 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.024

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{\left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      7. lower-cos.f64 62.5%

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if -0.024 < y < 0.068000000000000005

   1. Initial program 99.3%

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

   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 51.1%

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

   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 50.2%

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

   if 0.068000000000000005 < 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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 62.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.1%

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

Alternative 7: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.7639320225002103 \cdot \cos y\\ t_3 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.024:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot t\_3\right)\right), t\_0, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_2\right), 1.5, 3\right)}\\ \mathbf{elif}\;y \leq 0.068:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos x, \left(t\_0 \cdot \sqrt{2}\right) \cdot \left(\sin x + -0.0625 \cdot y\right), -2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_3 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), t\_0, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, 1, t\_2\right), 1.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.0625 (sin x) (sin y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* 0.7639320225002103 (cos y)))
        (t_3 (- 1.0 (cos y))))
   (if (<= y -0.024)
     (/
      (fma (* -0.0625 (* (sin y) (* (sqrt 2.0) t_3))) t_0 2.0)
      (fma (fma t_1 (cos x) t_2) 1.5 3.0))
     (if (<= y 0.068)
       (*
        (/
         (fma
          (- 1.0 (cos x))
          (* (* t_0 (sqrt 2.0)) (+ (sin x) (* -0.0625 y)))
          -2.0)
         (fma 0.5 (fma t_1 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 1.0))
        -0.3333333333333333)
       (/
        (fma (* (* t_3 (sqrt 2.0)) (fma (sin y) -0.0625 (sin x))) t_0 2.0)
        (fma (fma t_1 1.0 t_2) 1.5 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, sin(x), sin(y));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.7639320225002103 * cos(y);
	double t_3 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.024) {
		tmp = fma((-0.0625 * (sin(y) * (sqrt(2.0) * t_3))), t_0, 2.0) / fma(fma(t_1, cos(x), t_2), 1.5, 3.0);
	} else if (y <= 0.068) {
		tmp = (fma((1.0 - cos(x)), ((t_0 * sqrt(2.0)) * (sin(x) + (-0.0625 * y))), -2.0) / fma(0.5, fma(t_1, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0)) * -0.3333333333333333;
	} else {
		tmp = fma(((t_3 * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), t_0, 2.0) / fma(fma(t_1, 1.0, t_2), 1.5, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.0625, sin(x), sin(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.7639320225002103 * cos(y))
	t_3 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.024)
		tmp = Float64(fma(Float64(-0.0625 * Float64(sin(y) * Float64(sqrt(2.0) * t_3))), t_0, 2.0) / fma(fma(t_1, cos(x), t_2), 1.5, 3.0));
	elseif (y <= 0.068)
		tmp = Float64(Float64(fma(Float64(1.0 - cos(x)), Float64(Float64(t_0 * sqrt(2.0)) * Float64(sin(x) + Float64(-0.0625 * y))), -2.0) / fma(0.5, fma(t_1, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)) * -0.3333333333333333);
	else
		tmp = Float64(fma(Float64(Float64(t_3 * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), t_0, 2.0) / fma(fma(t_1, 1.0, t_2), 1.5, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.024], N[(N[(N[(-0.0625 * N[(N[Sin[y], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.068], N[(N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(N[(t$95$1 * 1.0 + t$95$2), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.024

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{\left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      7. lower-cos.f64 62.5%

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if -0.024 < y < 0.068000000000000005

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 62.5%

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

   8. Applied rewrites 62.5%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos x, \left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x + \frac{-1}{16} \cdot y\right)}, -2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot -0.3333333333333333 \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 57.3%

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

   11. Applied rewrites 57.3%

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

   if 0.068000000000000005 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(3 - \sqrt{5}\right) \cdot \cos y\\ t_3 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.024:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot t\_3\right)\right), t\_0, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)}\\ \mathbf{elif}\;y \leq 0.068:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos x, \left(t\_0 \cdot \sqrt{2}\right) \cdot \left(\sin x + -0.0625 \cdot y\right), -2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_2\right), 1\right)} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_3 \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\frac{\mathsf{fma}\left(t\_1, 1, t\_2\right)}{2} + 1}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.0625 (sin x) (sin y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (- 3.0 (sqrt 5.0)) (cos y)))
        (t_3 (- 1.0 (cos y))))
   (if (<= y -0.024)
     (/
      (fma (* -0.0625 (* (sin y) (* (sqrt 2.0) t_3))) t_0 2.0)
      (fma (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 1.5 3.0))
     (if (<= y 0.068)
       (*
        (/
         (fma
          (- 1.0 (cos x))
          (* (* t_0 (sqrt 2.0)) (+ (sin x) (* -0.0625 y)))
          -2.0)
         (fma 0.5 (fma t_1 (cos x) t_2) 1.0))
        -0.3333333333333333)
       (/
        (/
         (fma
          (* t_3 (sqrt 2.0))
          (* (fma (sin x) -0.0625 (sin y)) (fma (sin y) -0.0625 (sin x)))
          2.0)
         (+ (/ (fma t_1 1.0 t_2) 2.0) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, sin(x), sin(y));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (3.0 - sqrt(5.0)) * cos(y);
	double t_3 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.024) {
		tmp = fma((-0.0625 * (sin(y) * (sqrt(2.0) * t_3))), t_0, 2.0) / fma(fma(t_1, cos(x), (0.7639320225002103 * cos(y))), 1.5, 3.0);
	} else if (y <= 0.068) {
		tmp = (fma((1.0 - cos(x)), ((t_0 * sqrt(2.0)) * (sin(x) + (-0.0625 * y))), -2.0) / fma(0.5, fma(t_1, cos(x), t_2), 1.0)) * -0.3333333333333333;
	} else {
		tmp = (fma((t_3 * sqrt(2.0)), (fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / ((fma(t_1, 1.0, t_2) / 2.0) + 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.0625, sin(x), sin(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(3.0 - sqrt(5.0)) * cos(y))
	t_3 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.024)
		tmp = Float64(fma(Float64(-0.0625 * Float64(sin(y) * Float64(sqrt(2.0) * t_3))), t_0, 2.0) / fma(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))), 1.5, 3.0));
	elseif (y <= 0.068)
		tmp = Float64(Float64(fma(Float64(1.0 - cos(x)), Float64(Float64(t_0 * sqrt(2.0)) * Float64(sin(x) + Float64(-0.0625 * y))), -2.0) / fma(0.5, fma(t_1, cos(x), t_2), 1.0)) * -0.3333333333333333);
	else
		tmp = Float64(Float64(fma(Float64(t_3 * sqrt(2.0)), Float64(fma(sin(x), -0.0625, sin(y)) * fma(sin(y), -0.0625, sin(x))), 2.0) / Float64(Float64(fma(t_1, 1.0, t_2) / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.024], N[(N[(N[(-0.0625 * N[(N[Sin[y], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.068], N[(N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * 1.0 + t$95$2), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.024

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{\left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      7. lower-cos.f64 62.5%

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if -0.024 < y < 0.068000000000000005

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 62.5%

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

   8. Applied rewrites 62.5%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos x, \left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x + \frac{-1}{16} \cdot y\right)}, -2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot -0.3333333333333333 \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 57.3%

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

   11. Applied rewrites 57.3%

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

   if 0.068000000000000005 < 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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 62.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.1%

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

Alternative 9: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_3 := 0.7639320225002103 \cdot \cos y\\ t_4 := 1 - \cos y\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot t\_4\right)\right), t\_0, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_3\right), 1.5, 3\right)}\\ \mathbf{elif}\;y \leq 0.068:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos x, \left(t\_0 \cdot \sqrt{2}\right) \cdot t\_2, -2\right)}{1 + 0.5 \cdot \left(\left(3 + \cos x \cdot t\_1\right) - \sqrt{5}\right)} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_4 \cdot \sqrt{2}\right) \cdot t\_2, t\_0, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, 1, t\_3\right), 1.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.0625 (sin x) (sin y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (fma (sin y) -0.0625 (sin x)))
        (t_3 (* 0.7639320225002103 (cos y)))
        (t_4 (- 1.0 (cos y))))
   (if (<= y -1.8e+21)
     (/
      (fma (* -0.0625 (* (sin y) (* (sqrt 2.0) t_4))) t_0 2.0)
      (fma (fma t_1 (cos x) t_3) 1.5 3.0))
     (if (<= y 0.068)
       (*
        (/
         (fma (- 1.0 (cos x)) (* (* t_0 (sqrt 2.0)) t_2) -2.0)
         (+ 1.0 (* 0.5 (- (+ 3.0 (* (cos x) t_1)) (sqrt 5.0)))))
        -0.3333333333333333)
       (/
        (fma (* (* t_4 (sqrt 2.0)) t_2) t_0 2.0)
        (fma (fma t_1 1.0 t_3) 1.5 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, sin(x), sin(y));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma(sin(y), -0.0625, sin(x));
	double t_3 = 0.7639320225002103 * cos(y);
	double t_4 = 1.0 - cos(y);
	double tmp;
	if (y <= -1.8e+21) {
		tmp = fma((-0.0625 * (sin(y) * (sqrt(2.0) * t_4))), t_0, 2.0) / fma(fma(t_1, cos(x), t_3), 1.5, 3.0);
	} else if (y <= 0.068) {
		tmp = (fma((1.0 - cos(x)), ((t_0 * sqrt(2.0)) * t_2), -2.0) / (1.0 + (0.5 * ((3.0 + (cos(x) * t_1)) - sqrt(5.0))))) * -0.3333333333333333;
	} else {
		tmp = fma(((t_4 * sqrt(2.0)) * t_2), t_0, 2.0) / fma(fma(t_1, 1.0, t_3), 1.5, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.0625, sin(x), sin(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(sin(y), -0.0625, sin(x))
	t_3 = Float64(0.7639320225002103 * cos(y))
	t_4 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -1.8e+21)
		tmp = Float64(fma(Float64(-0.0625 * Float64(sin(y) * Float64(sqrt(2.0) * t_4))), t_0, 2.0) / fma(fma(t_1, cos(x), t_3), 1.5, 3.0));
	elseif (y <= 0.068)
		tmp = Float64(Float64(fma(Float64(1.0 - cos(x)), Float64(Float64(t_0 * sqrt(2.0)) * t_2), -2.0) / Float64(1.0 + Float64(0.5 * Float64(Float64(3.0 + Float64(cos(x) * t_1)) - sqrt(5.0))))) * -0.3333333333333333);
	else
		tmp = Float64(fma(Float64(Float64(t_4 * sqrt(2.0)) * t_2), t_0, 2.0) / fma(fma(t_1, 1.0, t_3), 1.5, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e+21], N[(N[(N[(-0.0625 * N[(N[Sin[y], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.068], N[(N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + -2.0), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(t$95$4 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(N[(t$95$1 * 1.0 + t$95$3), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e21

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{\left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      7. lower-cos.f64 62.5%

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if -1.8e21 < y < 0.068000000000000005

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 62.5%

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

   8. Applied rewrites 62.5%

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

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-sqrt.f64 60.4%

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

   11. Applied rewrites 60.4%

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

   if 0.068000000000000005 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.7639320225002103 \cdot \cos y\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_2\right), 1.5, 3\right)\\ t_4 := 1 - \cos y\\ \mathbf{if}\;y \leq -2500:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot t\_4\right)\right), t\_0, 2\right)}{t\_3}\\ \mathbf{elif}\;y \leq 0.068:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), t\_0, 2\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_4 \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), t\_0, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, 1, t\_2\right), 1.5, 3\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.0625 (sin x) (sin y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* 0.7639320225002103 (cos y)))
        (t_3 (fma (fma t_1 (cos x) t_2) 1.5 3.0))
        (t_4 (- 1.0 (cos y))))
   (if (<= y -2500.0)
     (/ (fma (* -0.0625 (* (sin y) (* (sqrt 2.0) t_4))) t_0 2.0) t_3)
     (if (<= y 0.068)
       (/ (fma (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0))) t_0 2.0) t_3)
       (/
        (fma (* (* t_4 (sqrt 2.0)) (fma (sin y) -0.0625 (sin x))) t_0 2.0)
        (fma (fma t_1 1.0 t_2) 1.5 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, sin(x), sin(y));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.7639320225002103 * cos(y);
	double t_3 = fma(fma(t_1, cos(x), t_2), 1.5, 3.0);
	double t_4 = 1.0 - cos(y);
	double tmp;
	if (y <= -2500.0) {
		tmp = fma((-0.0625 * (sin(y) * (sqrt(2.0) * t_4))), t_0, 2.0) / t_3;
	} else if (y <= 0.068) {
		tmp = fma((sin(x) * (sqrt(2.0) * (cos(x) - 1.0))), t_0, 2.0) / t_3;
	} else {
		tmp = fma(((t_4 * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), t_0, 2.0) / fma(fma(t_1, 1.0, t_2), 1.5, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.0625, sin(x), sin(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.7639320225002103 * cos(y))
	t_3 = fma(fma(t_1, cos(x), t_2), 1.5, 3.0)
	t_4 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -2500.0)
		tmp = Float64(fma(Float64(-0.0625 * Float64(sin(y) * Float64(sqrt(2.0) * t_4))), t_0, 2.0) / t_3);
	elseif (y <= 0.068)
		tmp = Float64(fma(Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))), t_0, 2.0) / t_3);
	else
		tmp = Float64(fma(Float64(Float64(t_4 * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))), t_0, 2.0) / fma(fma(t_1, 1.0, t_2), 1.5, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2500.0], N[(N[(N[(-0.0625 * N[(N[Sin[y], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 0.068], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[(t$95$4 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(N[(t$95$1 * 1.0 + t$95$2), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2500

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{\left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      7. lower-cos.f64 62.5%

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if -2500 < y < 0.068000000000000005

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower-cos.f64 62.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if 0.068000000000000005 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\color{blue}{1} - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 60.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{1}, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)\\ \mathbf{if}\;y \leq -2500:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right), t\_0, 2\right)}{t\_2}\\ \mathbf{elif}\;y \leq 0.043:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), t\_0, 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\frac{\mathsf{fma}\left(t\_1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.0625 (sin x) (sin y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (fma (fma t_1 (cos x) (* 0.7639320225002103 (cos y))) 1.5 3.0)))
   (if (<= y -2500.0)
     (/
      (fma (* -0.0625 (* (sin y) (* (sqrt 2.0) (- 1.0 (cos y))))) t_0 2.0)
      t_2)
     (if (<= y 0.043)
       (/ (fma (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0))) t_0 2.0) t_2)
       (/
        (/
         (fma
          (* (- (cos x) (cos y)) (sqrt 2.0))
          (* -0.0625 (pow (sin y) 2.0))
          2.0)
         (+ (/ (fma t_1 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 2.0) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, sin(x), sin(y));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma(fma(t_1, cos(x), (0.7639320225002103 * cos(y))), 1.5, 3.0);
	double tmp;
	if (y <= -2500.0) {
		tmp = fma((-0.0625 * (sin(y) * (sqrt(2.0) * (1.0 - cos(y))))), t_0, 2.0) / t_2;
	} else if (y <= 0.043) {
		tmp = fma((sin(x) * (sqrt(2.0) * (cos(x) - 1.0))), t_0, 2.0) / t_2;
	} else {
		tmp = (fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * pow(sin(y), 2.0)), 2.0) / ((fma(t_1, cos(x), ((3.0 - sqrt(5.0)) * cos(y))) / 2.0) + 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.0625, sin(x), sin(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(fma(t_1, cos(x), Float64(0.7639320225002103 * cos(y))), 1.5, 3.0)
	tmp = 0.0
	if (y <= -2500.0)
		tmp = Float64(fma(Float64(-0.0625 * Float64(sin(y) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))), t_0, 2.0) / t_2);
	elseif (y <= 0.043)
		tmp = Float64(fma(Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))), t_0, 2.0) / t_2);
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / Float64(Float64(fma(t_1, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]}, If[LessEqual[y, -2500.0], N[(N[(N[(-0.0625 * N[(N[Sin[y], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y, 0.043], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2500

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{\left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      7. lower-cos.f64 62.5%

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-0.0625 \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if -2500 < y < 0.042999999999999997

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower-cos.f64 62.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if 0.042999999999999997 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.5%

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

   6. Applied rewrites 62.5%

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

Alternative 12: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)\\ t_2 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -2500:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t\_1}\\ \mathbf{elif}\;y \leq 0.043:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot t\_2, 2\right)}{\frac{\mathsf{fma}\left(t\_0, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (fma (fma t_0 (cos x) (* 0.7639320225002103 (cos y))) 1.5 3.0))
        (t_2 (pow (sin y) 2.0)))
   (if (<= y -2500.0)
     (/ (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) (- 1.0 (cos y)))))) t_1)
     (if (<= y 0.043)
       (/
        (fma
         (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0)))
         (fma -0.0625 (sin x) (sin y))
         2.0)
        t_1)
       (/
        (/
         (fma (* (- (cos x) (cos y)) (sqrt 2.0)) (* -0.0625 t_2) 2.0)
         (+ (/ (fma t_0 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 2.0) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma(fma(t_0, cos(x), (0.7639320225002103 * cos(y))), 1.5, 3.0);
	double t_2 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -2500.0) {
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(y)))))) / t_1;
	} else if (y <= 0.043) {
		tmp = fma((sin(x) * (sqrt(2.0) * (cos(x) - 1.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / t_1;
	} else {
		tmp = (fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * t_2), 2.0) / ((fma(t_0, cos(x), ((3.0 - sqrt(5.0)) * cos(y))) / 2.0) + 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(fma(t_0, cos(x), Float64(0.7639320225002103 * cos(y))), 1.5, 3.0)
	t_2 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -2500.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_1);
	elseif (y <= 0.043)
		tmp = Float64(fma(Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))), fma(-0.0625, sin(x), sin(y)), 2.0) / t_1);
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * t_2), 2.0) / Float64(Float64(fma(t_0, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -2500.0], 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] / t$95$1), $MachinePrecision], If[LessEqual[y, 0.043], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2500

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      9. lower-cos.f64 62.5%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if -2500 < y < 0.042999999999999997

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right), \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower-cos.f64 62.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if 0.042999999999999997 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.5%

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

   6. Applied rewrites 62.5%

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

Alternative 13: 78.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)}\\ \mathbf{elif}\;y \leq 0.068:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, -0.0625 \cdot t\_1, 2\right)}{\frac{\mathsf{fma}\left(t\_0, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right)}{2} + 1}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (pow (sin y) 2.0)))
   (if (<= y -1.8e+21)
     (/
      (+ 2.0 (* -0.0625 (* t_1 (* (sqrt 2.0) (- 1.0 (cos y))))))
      (fma (fma t_0 (cos x) (* 0.7639320225002103 (cos y))) 1.5 3.0))
     (if (<= y 0.068)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (+ 3.0 (* 1.5 (+ 0.7639320225002103 (* (cos x) t_0)))))
       (/
        (/
         (fma (* (- (cos x) (cos y)) (sqrt 2.0)) (* -0.0625 t_1) 2.0)
         (+ (/ (fma t_0 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 2.0) 1.0))
        3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -1.8e+21) {
		tmp = (2.0 + (-0.0625 * (t_1 * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(fma(t_0, cos(x), (0.7639320225002103 * cos(y))), 1.5, 3.0);
	} else if (y <= 0.068) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (1.5 * (0.7639320225002103 + (cos(x) * t_0))));
	} else {
		tmp = (fma(((cos(x) - cos(y)) * sqrt(2.0)), (-0.0625 * t_1), 2.0) / ((fma(t_0, cos(x), ((3.0 - sqrt(5.0)) * cos(y))) / 2.0) + 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -1.8e+21)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_1 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(fma(t_0, cos(x), Float64(0.7639320225002103 * cos(y))), 1.5, 3.0));
	elseif (y <= 0.068)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + Float64(1.5 * Float64(0.7639320225002103 + Float64(cos(x) * t_0)))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), Float64(-0.0625 * t_1), 2.0) / Float64(Float64(fma(t_0, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))) / 2.0) + 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -1.8e+21], N[(N[(2.0 + N[(-0.0625 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.068], N[(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] / N[(3.0 + N[(1.5 * N[(0.7639320225002103 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8e21

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      9. lower-cos.f64 62.5%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if -1.8e21 < y < 0.068000000000000005

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   9. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   if 0.068000000000000005 < 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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.5%

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

   6. Applied rewrites 62.5%

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

Alternative 14: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.068:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
          (fma (fma t_0 (cos x) (* 0.7639320225002103 (cos y))) 1.5 3.0))))
   (if (<= y -1.8e+21)
     t_1
     (if (<= y 0.068)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (+ 3.0 (* 1.5 (+ 0.7639320225002103 (* (cos x) t_0)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(fma(t_0, cos(x), (0.7639320225002103 * cos(y))), 1.5, 3.0);
	double tmp;
	if (y <= -1.8e+21) {
		tmp = t_1;
	} else if (y <= 0.068) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (1.5 * (0.7639320225002103 + (cos(x) * t_0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(fma(t_0, cos(x), Float64(0.7639320225002103 * cos(y))), 1.5, 3.0))
	tmp = 0.0
	if (y <= -1.8e+21)
		tmp = t_1;
	elseif (y <= 0.068)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + Float64(1.5 * Float64(0.7639320225002103 + Float64(cos(x) * t_0)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(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[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(0.7639320225002103 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e+21], t$95$1, If[LessEqual[y, 0.068], N[(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] / N[(3.0 + N[(1.5 * N[(0.7639320225002103 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e21 or 0.068000000000000005 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{6880887943736673}{9007199254740992} \cdot \cos y\right), \frac{3}{2}, 3\right)} \]

      9. lower-cos.f64 62.5%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   9. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 0.7639320225002103 \cdot \cos y\right), 1.5, 3\right)} \]

   if -1.8e21 < y < 0.068000000000000005

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   9. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\mathsf{fma}\left(0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot \frac{1}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2\right), -1\right)}\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\left(1 + \frac{\mathsf{fma}\left(t\_2, \cos y, -1\right)}{\sqrt{5}}\right) \cdot \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -3.2e-7)
     (*
      0.3333333333333333
      (*
       (fma (* 0.0625 (* t_1 (sqrt 2.0))) (- 0.5 (* 0.5 (cos (* 2.0 x)))) -2.0)
       (/ 1.0 (fma -0.5 (fma t_0 (cos x) t_2) -1.0))))
     (if (<= x 3e+15)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+
         3.0
         (* 1.5 (* (+ 1.0 (/ (fma t_2 (cos y) -1.0) (sqrt 5.0))) (sqrt 5.0)))))
       (/
        (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_1))))
        (+ 3.0 (* 1.5 (+ 0.7639320225002103 (* (cos x) t_0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -3.2e-7) {
		tmp = 0.3333333333333333 * (fma((0.0625 * (t_1 * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), -2.0) * (1.0 / fma(-0.5, fma(t_0, cos(x), t_2), -1.0)));
	} else if (x <= 3e+15) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((1.0 + (fma(t_2, cos(y), -1.0) / sqrt(5.0))) * sqrt(5.0))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_1)))) / (3.0 + (1.5 * (0.7639320225002103 + (cos(x) * t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -3.2e-7)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(0.0625 * Float64(t_1 * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), -2.0) * Float64(1.0 / fma(-0.5, fma(t_0, cos(x), t_2), -1.0))));
	elseif (x <= 3e+15)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(1.0 + Float64(fma(t_2, cos(y), -1.0) / sqrt(5.0))) * sqrt(5.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_1)))) / Float64(3.0 + Float64(1.5 * Float64(0.7639320225002103 + Float64(cos(x) * t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-7], N[(0.3333333333333333 * N[(N[(N[(0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * N[(1.0 / N[(-0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+15], 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[(3.0 + N[(1.5 * N[(N[(1.0 + N[(N[(t$95$2 * N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(0.7639320225002103 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-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

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   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 59.9%

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

   6. Applied rewrites 59.9%

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

   if -3.2000000000000001e-7 < x < 3e15

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sub-flip N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 59.6%

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

   10. Applied rewrites 59.6%

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

   if 3e15 < 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%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   9. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\mathsf{fma}\left(0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot \frac{1}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2\right), -1\right)}\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + 0.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_2\right) - 1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -3.2e-7)
     (*
      0.3333333333333333
      (*
       (fma (* 0.0625 (* t_1 (sqrt 2.0))) (- 0.5 (* 0.5 (cos (* 2.0 x)))) -2.0)
       (/ 1.0 (fma -0.5 (fma t_0 (cos x) t_2) -1.0))))
     (if (<= x 3e+15)
       (/
        (/
         (+
          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_2)) 1.0))))
        3.0)
       (/
        (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_1))))
        (+ 3.0 (* 1.5 (+ 0.7639320225002103 (* (cos x) t_0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -3.2e-7) {
		tmp = 0.3333333333333333 * (fma((0.0625 * (t_1 * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), -2.0) * (1.0 / fma(-0.5, fma(t_0, cos(x), t_2), -1.0)));
	} else if (x <= 3e+15) {
		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_2)) - 1.0)))) / 3.0;
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_1)))) / (3.0 + (1.5 * (0.7639320225002103 + (cos(x) * t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -3.2e-7)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(0.0625 * Float64(t_1 * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), -2.0) * Float64(1.0 / fma(-0.5, fma(t_0, cos(x), t_2), -1.0))));
	elseif (x <= 3e+15)
		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_2)) - 1.0)))) / 3.0);
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_1)))) / Float64(3.0 + Float64(1.5 * Float64(0.7639320225002103 + Float64(cos(x) * t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-7], N[(0.3333333333333333 * N[(N[(N[(0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * N[(1.0 / N[(-0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+15], 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$2), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(0.7639320225002103 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-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

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   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 59.9%

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

   6. Applied rewrites 59.9%

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

   if -3.2000000000000001e-7 < x < 3e15

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 59.6%

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

   if 3e15 < 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%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   9. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 78.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\mathsf{fma}\left(0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot \frac{1}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2\right), -1\right)}\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_2\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -3.2e-7)
     (*
      0.3333333333333333
      (*
       (fma (* 0.0625 (* t_1 (sqrt 2.0))) (- 0.5 (* 0.5 (cos (* 2.0 x)))) -2.0)
       (/ 1.0 (fma -0.5 (fma t_0 (cos x) t_2) -1.0))))
     (if (<= x 3e+15)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* (cos y) t_2)) 1.0))))
       (/
        (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_1))))
        (+ 3.0 (* 1.5 (+ 0.7639320225002103 (* (cos x) t_0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -3.2e-7) {
		tmp = 0.3333333333333333 * (fma((0.0625 * (t_1 * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), -2.0) * (1.0 / fma(-0.5, fma(t_0, cos(x), t_2), -1.0)));
	} else if (x <= 3e+15) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 + (1.5 * ((sqrt(5.0) + (cos(y) * t_2)) - 1.0)));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_1)))) / (3.0 + (1.5 * (0.7639320225002103 + (cos(x) * t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -3.2e-7)
		tmp = Float64(0.3333333333333333 * Float64(fma(Float64(0.0625 * Float64(t_1 * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), -2.0) * Float64(1.0 / fma(-0.5, fma(t_0, cos(x), t_2), -1.0))));
	elseif (x <= 3e+15)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * t_2)) - 1.0))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_1)))) / Float64(3.0 + Float64(1.5 * Float64(0.7639320225002103 + Float64(cos(x) * t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-7], N[(0.3333333333333333 * N[(N[(N[(0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * N[(1.0 / N[(-0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+15], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(0.7639320225002103 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-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

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   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 59.9%

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

   6. Applied rewrites 59.9%

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

   if -3.2000000000000001e-7 < x < 3e15

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 59.5%

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

   if 3e15 < 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%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   9. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 78.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2\right), -1\right)}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_2\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -3.2e-7)
     (/
      (*
       (fma (* 0.0625 (* t_1 (sqrt 2.0))) (- 0.5 (* 0.5 (cos (* 2.0 x)))) -2.0)
       0.3333333333333333)
      (fma -0.5 (fma t_0 (cos x) t_2) -1.0))
     (if (<= x 3e+15)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* (cos y) t_2)) 1.0))))
       (/
        (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) t_1))))
        (+ 3.0 (* 1.5 (+ 0.7639320225002103 (* (cos x) t_0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -3.2e-7) {
		tmp = (fma((0.0625 * (t_1 * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma(t_0, cos(x), t_2), -1.0);
	} else if (x <= 3e+15) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 + (1.5 * ((sqrt(5.0) + (cos(y) * t_2)) - 1.0)));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * t_1)))) / (3.0 + (1.5 * (0.7639320225002103 + (cos(x) * t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -3.2e-7)
		tmp = Float64(Float64(fma(Float64(0.0625 * Float64(t_1 * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), -2.0) * 0.3333333333333333) / fma(-0.5, fma(t_0, cos(x), t_2), -1.0));
	elseif (x <= 3e+15)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * t_2)) - 1.0))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * t_1)))) / Float64(3.0 + Float64(1.5 * Float64(0.7639320225002103 + Float64(cos(x) * t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-7], N[(N[(N[(N[(0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+15], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(0.7639320225002103 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-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

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   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 59.9%

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

   6. Applied rewrites 59.9%

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

   if -3.2000000000000001e-7 < x < 3e15

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 59.5%

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

   if 3e15 < 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%

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

   5. Applied rewrites 99.3%

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

   6. Evaluated real constant 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{0.7639320225002103} \cdot \cos y\right), 1.5, 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{3 + \frac{3}{2} \cdot \left(\frac{6880887943736673}{9007199254740992} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   9. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(0.7639320225002103 + \cos x \cdot \left(\sqrt{5} - 1\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_2\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625 \cdot t\_1, t\_0, -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, t\_3, -1\right)}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_2\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, t\_0, 2\right)}{\mathsf{fma}\left(t\_3, 0.5, 1\right)}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma (- (sqrt 5.0) 1.0) (cos x) t_2)))
   (if (<= x -3.2e-7)
     (/
      (* (fma (* 0.0625 t_1) t_0 -2.0) 0.3333333333333333)
      (fma -0.5 t_3 -1.0))
     (if (<= x 3e+15)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* (cos y) t_2)) 1.0))))
       (/ (/ (fma (* -0.0625 t_1) t_0 2.0) (fma t_3 0.5 1.0)) 3.0)))))

C

double code(double x, double y) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * x)));
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma((sqrt(5.0) - 1.0), cos(x), t_2);
	double tmp;
	if (x <= -3.2e-7) {
		tmp = (fma((0.0625 * t_1), t_0, -2.0) * 0.3333333333333333) / fma(-0.5, t_3, -1.0);
	} else if (x <= 3e+15) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 + (1.5 * ((sqrt(5.0) + (cos(y) * t_2)) - 1.0)));
	} else {
		tmp = (fma((-0.0625 * t_1), t_0, 2.0) / fma(t_3, 0.5, 1.0)) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(Float64(sqrt(5.0) - 1.0), cos(x), t_2)
	tmp = 0.0
	if (x <= -3.2e-7)
		tmp = Float64(Float64(fma(Float64(0.0625 * t_1), t_0, -2.0) * 0.3333333333333333) / fma(-0.5, t_3, -1.0));
	elseif (x <= 3e+15)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * t_2)) - 1.0))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_1), t_0, 2.0) / fma(t_3, 0.5, 1.0)) / 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[x, -3.2e-7], N[(N[(N[(N[(0.0625 * t$95$1), $MachinePrecision] * t$95$0 + -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+15], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(t$95$3 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-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

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   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 59.9%

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

   6. Applied rewrites 59.9%

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

   if -3.2000000000000001e-7 < x < 3e15

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 59.5%

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

   if 3e15 < 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 59.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 59.9%

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

   6. Applied rewrites 59.9%

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

Alternative 20: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_2\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625 \cdot t\_1, t\_0, -2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(-0.5, t\_3, -1\right)}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_2\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, t\_0, 2\right)}{\mathsf{fma}\left(t\_3, 0.5, 1\right)} \cdot 0.3333333333333333\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma (- (sqrt 5.0) 1.0) (cos x) t_2)))
   (if (<= x -3.2e-7)
     (/
      (* (fma (* 0.0625 t_1) t_0 -2.0) 0.3333333333333333)
      (fma -0.5 t_3 -1.0))
     (if (<= x 3e+15)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* (cos y) t_2)) 1.0))))
       (*
        (/ (fma (* -0.0625 t_1) t_0 2.0) (fma t_3 0.5 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * x)));
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma((sqrt(5.0) - 1.0), cos(x), t_2);
	double tmp;
	if (x <= -3.2e-7) {
		tmp = (fma((0.0625 * t_1), t_0, -2.0) * 0.3333333333333333) / fma(-0.5, t_3, -1.0);
	} else if (x <= 3e+15) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 + (1.5 * ((sqrt(5.0) + (cos(y) * t_2)) - 1.0)));
	} else {
		tmp = (fma((-0.0625 * t_1), t_0, 2.0) / fma(t_3, 0.5, 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(Float64(sqrt(5.0) - 1.0), cos(x), t_2)
	tmp = 0.0
	if (x <= -3.2e-7)
		tmp = Float64(Float64(fma(Float64(0.0625 * t_1), t_0, -2.0) * 0.3333333333333333) / fma(-0.5, t_3, -1.0));
	elseif (x <= 3e+15)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * t_2)) - 1.0))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_1), t_0, 2.0) / fma(t_3, 0.5, 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[x, -3.2e-7], N[(N[(N[(N[(0.0625 * t$95$1), $MachinePrecision] * t$95$0 + -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(-0.5 * t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+15], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(t$95$3 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-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

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   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 59.9%

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

   6. Applied rewrites 59.9%

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

   if -3.2000000000000001e-7 < x < 3e15

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 59.5%

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

   if 3e15 < 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 59.9%

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

   6. Applied rewrites 59.9%

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

Alternative 21: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right), 0.5, 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + \cos y \cdot t\_0\right) - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
            (- 0.5 (* 0.5 (cos (* 2.0 x))))
            2.0)
           (fma (fma (- (sqrt 5.0) 1.0) (cos x) t_0) 0.5 1.0))
          0.3333333333333333)))
   (if (<= x -3.2e-7)
     t_1
     (if (<= x 3e+15)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (+ 3.0 (* 1.5 (- (+ (sqrt 5.0) (* (cos y) t_0)) 1.0))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (fma((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))), (0.5 - (0.5 * cos((2.0 * x)))), 2.0) / fma(fma((sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -3.2e-7) {
		tmp = t_1;
	} else if (x <= 3e+15) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 + (1.5 * ((sqrt(5.0) + (cos(y) * t_0)) - 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(fma(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))), 2.0) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), t_0), 0.5, 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -3.2e-7)
		tmp = t_1;
	elseif (x <= 3e+15)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + Float64(cos(y) * t_0)) - 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -3.2e-7], t$95$1, If[LessEqual[x, 3e+15], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000001e-7 or 3e15 < 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 59.9%

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

   6. Applied rewrites 59.9%

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

   if -3.2000000000000001e-7 < x < 3e15

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 59.5%

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

Alternative 22: 59.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

8. Applied rewrites 59.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

10. Applied rewrites 59.5%

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

Alternative 23: 59.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

8. Applied rewrites 59.6%

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

10. Applied rewrites 59.5%

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

Alternative 24: 43.0% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(2.0 / N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] + 1.5), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $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 59.9%

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

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

   11. lower--.f64 N/A

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

6. Applied rewrites 59.9%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 43.0%

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

Alternative 25: 40.5% accurate, 316.7× speedup

Math

\[0.3333333333333333 \]

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 59.9%

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

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

   11. lower--.f64 N/A

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

6. Applied rewrites 59.9%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 40.5%

   \[\leadsto 0.3333333333333333 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(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))))))