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

Percentage Accurate: 99.3% → 99.3%

Time: 57.3s

Alternatives: 23

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}} + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
   (* (- (sin y) (/ (sin x) 16.0)) (- (cos x) (cos y)))
   2.0)
  (*
   3.0
   (+
    1.0
    (+
     (/ (cos y) (+ 1.5 (* 0.5 (sqrt 5.0))))
     (* 0.5 (* (cos x) (+ (sqrt 5.0) -1.0))))))))

C

double code(double x, double y) {
	return fma((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))), ((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y))), 2.0) / (3.0 * (1.0 + ((cos(y) / (1.5 + (0.5 * sqrt(5.0)))) + (0.5 * (cos(x) * (sqrt(5.0) + -1.0))))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y))), 2.0) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(y) / Float64(1.5 + Float64(0.5 * sqrt(5.0)))) + Float64(0.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))))))
end

Wolfram

code[x_, y_] := 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[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[y], $MachinePrecision] / N[(1.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. cos-neg 99.4%

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

   6. distribute-lft-in 99.4%

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

3. Simplified 99.4%

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

   1. flip-- 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \color{blue}{\frac{1.5 \cdot 1.5 - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   2. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   3. div-inv 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   4. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   5. div-inv 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   7. div-inv 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   8. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]
6. Step-by-step derivation

   1. swap-sqr 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   2. rem-square-sqrt 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   3. cancel-sign-sub-inv 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   4. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   7. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{\color{blue}{1}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   8. +-commutative 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{1}{\color{blue}{\sqrt{5} \cdot 0.5 + 1.5}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   9. *-commutative 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{1}{\color{blue}{0.5 \cdot \sqrt{5}} + 1.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   10. fma-def 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

7. Simplified 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

8. Taylor expanded in y around inf 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}}\right)\right)}} \]

9. Final simplification 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}} + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)\right)} \]

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (- (cos x) (cos y))
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))))
  (*
   3.0
   (+
    (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
    (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0))))))

C

double code(double x, double y) {
	return (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
}

Python

def code(x, y):
	return (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. flip-- 99.3%

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

   2. metadata-eval 99.3%

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

   3. add-sqr-sqrt 99.4%

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

   4. metadata-eval 99.4%

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

3. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (/
    (+
     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 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	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 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = sqrt(5.0d0) / 2.0d0
    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 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	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.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	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.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	return Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	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 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.4%

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

   2. distribute-lft-in 99.4%

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

   3. cos-neg 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. associate-+l+ 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}} \]

4. Final simplification 99.4%

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)} \]

Alternative 4: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0115 \lor \neg \left(x \leq 0.042\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \sin x, t_0 \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}} + 0.5 \cdot \left(\cos x \cdot t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_0\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -0.0115) (not (<= x 0.042)))
     (/
      (fma (* (sqrt 2.0) (sin x)) (* t_0 (- (cos x) (cos y))) 2.0)
      (*
       3.0
       (+
        1.0
        (+ (/ (cos y) (+ 1.5 (* 0.5 (sqrt 5.0)))) (* 0.5 (* (cos x) t_1))))))
     (/
      (+
       2.0
       (*
        (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_0)
        (+ 1.0 (- (* -0.5 (* x x)) (cos y)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_1 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.0115) || !(x <= 0.042)) {
		tmp = fma((sqrt(2.0) * sin(x)), (t_0 * (cos(x) - cos(y))), 2.0) / (3.0 * (1.0 + ((cos(y) / (1.5 + (0.5 * sqrt(5.0)))) + (0.5 * (cos(x) * t_1)))));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_0) * (1.0 + ((-0.5 * (x * x)) - cos(y))))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -0.0115) || !(x <= 0.042))
		tmp = Float64(fma(Float64(sqrt(2.0) * sin(x)), Float64(t_0 * Float64(cos(x) - cos(y))), 2.0) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(y) / Float64(1.5 + Float64(0.5 * sqrt(5.0)))) + Float64(0.5 * Float64(cos(x) * t_1))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_0) * Float64(1.0 + Float64(Float64(-0.5 * Float64(x * x)) - cos(y))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0115], N[Not[LessEqual[x, 0.042]], $MachinePrecision]], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[y], $MachinePrecision] / N[(1.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0115 or 0.0420000000000000026 < x

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      2. associate-*l* 99.0%

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

      3. fma-def 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. cos-neg 99.0%

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

      6. distribute-lft-in 99.0%

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

   3. Simplified 99.0%

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

      1. flip-- 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \color{blue}{\frac{1.5 \cdot 1.5 - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      2. metadata-eval 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      3. div-inv 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      4. metadata-eval 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      5. div-inv 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      6. metadata-eval 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      7. div-inv 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      8. metadata-eval 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   5. Applied egg-rr 98.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      2. rem-square-sqrt 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      3. cancel-sign-sub-inv 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      4. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      5. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      6. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      7. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{\color{blue}{1}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      8. +-commutative 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{1}{\color{blue}{\sqrt{5} \cdot 0.5 + 1.5}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      9. *-commutative 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{1}{\color{blue}{0.5 \cdot \sqrt{5}} + 1.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      10. fma-def 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   7. Simplified 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   8. Taylor expanded in y around inf 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}}\right)\right)}} \]

   9. Taylor expanded in y around 0 61.1%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot \sin x}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}}\right)\right)} \]

   if -0.0115 < x < 0.0420000000000000026

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. associate--l+ 99.4%

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

      2. unpow2 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0115 \lor \neg \left(x \leq 0.042\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \sin x, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}} + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 5: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \cos x - \cos y\\ t_3 := \frac{\sqrt{5}}{2}\\ t_4 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0115:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_4 \cdot t_0\right)}{t_1}\\ \mathbf{elif}\;x \leq 0.049:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_4\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_4 \cdot t_2\right) \cdot t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_3 - 0.5\right) + \cos y \cdot \left(1.5 - t_3\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (sin x)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
        (t_2 (- (cos x) (cos y)))
        (t_3 (/ (sqrt 5.0) 2.0))
        (t_4 (- (sin y) (/ (sin x) 16.0))))
   (if (<= x -0.0115)
     (/ (+ 2.0 (* t_2 (* t_4 t_0))) t_1)
     (if (<= x 0.049)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_4)
          (+ 1.0 (- (* -0.5 (* x x)) (cos y)))))
        t_1)
       (/
        (+ 2.0 (* (* t_4 t_2) t_0))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_3 0.5)) (* (cos y) (- 1.5 t_3))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * sin(x);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double t_2 = cos(x) - cos(y);
	double t_3 = sqrt(5.0) / 2.0;
	double t_4 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if (x <= -0.0115) {
		tmp = (2.0 + (t_2 * (t_4 * t_0))) / t_1;
	} else if (x <= 0.049) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_4) * (1.0 + ((-0.5 * (x * x)) - cos(y))))) / t_1;
	} else {
		tmp = (2.0 + ((t_4 * t_2) * t_0)) / (3.0 * (1.0 + ((cos(x) * (t_3 - 0.5)) + (cos(y) * (1.5 - t_3)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * sin(x)
    t_1 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    t_2 = cos(x) - cos(y)
    t_3 = sqrt(5.0d0) / 2.0d0
    t_4 = sin(y) - (sin(x) / 16.0d0)
    if (x <= (-0.0115d0)) then
        tmp = (2.0d0 + (t_2 * (t_4 * t_0))) / t_1
    else if (x <= 0.049d0) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * t_4) * (1.0d0 + (((-0.5d0) * (x * x)) - cos(y))))) / t_1
    else
        tmp = (2.0d0 + ((t_4 * t_2) * t_0)) / (3.0d0 * (1.0d0 + ((cos(x) * (t_3 - 0.5d0)) + (cos(y) * (1.5d0 - t_3)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * Math.sin(x);
	double t_1 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double t_2 = Math.cos(x) - Math.cos(y);
	double t_3 = Math.sqrt(5.0) / 2.0;
	double t_4 = Math.sin(y) - (Math.sin(x) / 16.0);
	double tmp;
	if (x <= -0.0115) {
		tmp = (2.0 + (t_2 * (t_4 * t_0))) / t_1;
	} else if (x <= 0.049) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * t_4) * (1.0 + ((-0.5 * (x * x)) - Math.cos(y))))) / t_1;
	} else {
		tmp = (2.0 + ((t_4 * t_2) * t_0)) / (3.0 * (1.0 + ((Math.cos(x) * (t_3 - 0.5)) + (Math.cos(y) * (1.5 - t_3)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * math.sin(x)
	t_1 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	t_2 = math.cos(x) - math.cos(y)
	t_3 = math.sqrt(5.0) / 2.0
	t_4 = math.sin(y) - (math.sin(x) / 16.0)
	tmp = 0
	if x <= -0.0115:
		tmp = (2.0 + (t_2 * (t_4 * t_0))) / t_1
	elif x <= 0.049:
		tmp = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * t_4) * (1.0 + ((-0.5 * (x * x)) - math.cos(y))))) / t_1
	else:
		tmp = (2.0 + ((t_4 * t_2) * t_0)) / (3.0 * (1.0 + ((math.cos(x) * (t_3 - 0.5)) + (math.cos(y) * (1.5 - t_3)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * sin(x))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(sqrt(5.0) / 2.0)
	t_4 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if (x <= -0.0115)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_4 * t_0))) / t_1);
	elseif (x <= 0.049)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_4) * Float64(1.0 + Float64(Float64(-0.5 * Float64(x * x)) - cos(y))))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_4 * t_2) * t_0)) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_3 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_3))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * sin(x);
	t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	t_2 = cos(x) - cos(y);
	t_3 = sqrt(5.0) / 2.0;
	t_4 = sin(y) - (sin(x) / 16.0);
	tmp = 0.0;
	if (x <= -0.0115)
		tmp = (2.0 + (t_2 * (t_4 * t_0))) / t_1;
	elseif (x <= 0.049)
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_4) * (1.0 + ((-0.5 * (x * x)) - cos(y))))) / t_1;
	else
		tmp = (2.0 + ((t_4 * t_2) * t_0)) / (3.0 * (1.0 + ((cos(x) * (t_3 - 0.5)) + (cos(y) * (1.5 - t_3)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0115], N[(N[(2.0 + N[(t$95$2 * N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 0.049], 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] * t$95$4), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$4 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$3 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0115

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 56.9%

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

   if -0.0115 < x < 0.049000000000000002

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. associate--l+ 99.4%

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

      2. unpow2 99.4%

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

   4. Simplified 99.4%

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

   if 0.049000000000000002 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}} \]

   4. Taylor expanded in y around 0 64.7%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0115:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.049:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 6: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \frac{\sqrt{5}}{2}\\ t_3 := \cos x - \cos y\\ t_4 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0036:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_4 \cdot t_0\right)}{t_1}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_4\right) \cdot \left(1 - \cos y\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_4 \cdot t_3\right) \cdot t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_2 - 0.5\right) + \cos y \cdot \left(1.5 - t_2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (sin x)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
        (t_2 (/ (sqrt 5.0) 2.0))
        (t_3 (- (cos x) (cos y)))
        (t_4 (- (sin y) (/ (sin x) 16.0))))
   (if (<= x -0.0036)
     (/ (+ 2.0 (* t_3 (* t_4 t_0))) t_1)
     (if (<= x 6.5e+20)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_4)
          (- 1.0 (cos y))))
        t_1)
       (/
        (+ 2.0 (* (* t_4 t_3) t_0))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_2 0.5)) (* (cos y) (- 1.5 t_2))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * sin(x);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double t_2 = sqrt(5.0) / 2.0;
	double t_3 = cos(x) - cos(y);
	double t_4 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if (x <= -0.0036) {
		tmp = (2.0 + (t_3 * (t_4 * t_0))) / t_1;
	} else if (x <= 6.5e+20) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_4) * (1.0 - cos(y)))) / t_1;
	} else {
		tmp = (2.0 + ((t_4 * t_3) * t_0)) / (3.0 * (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * sin(x)
    t_1 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    t_2 = sqrt(5.0d0) / 2.0d0
    t_3 = cos(x) - cos(y)
    t_4 = sin(y) - (sin(x) / 16.0d0)
    if (x <= (-0.0036d0)) then
        tmp = (2.0d0 + (t_3 * (t_4 * t_0))) / t_1
    else if (x <= 6.5d+20) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * t_4) * (1.0d0 - cos(y)))) / t_1
    else
        tmp = (2.0d0 + ((t_4 * t_3) * t_0)) / (3.0d0 * (1.0d0 + ((cos(x) * (t_2 - 0.5d0)) + (cos(y) * (1.5d0 - t_2)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * Math.sin(x);
	double t_1 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double t_2 = Math.sqrt(5.0) / 2.0;
	double t_3 = Math.cos(x) - Math.cos(y);
	double t_4 = Math.sin(y) - (Math.sin(x) / 16.0);
	double tmp;
	if (x <= -0.0036) {
		tmp = (2.0 + (t_3 * (t_4 * t_0))) / t_1;
	} else if (x <= 6.5e+20) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * t_4) * (1.0 - Math.cos(y)))) / t_1;
	} else {
		tmp = (2.0 + ((t_4 * t_3) * t_0)) / (3.0 * (1.0 + ((Math.cos(x) * (t_2 - 0.5)) + (Math.cos(y) * (1.5 - t_2)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * math.sin(x)
	t_1 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	t_2 = math.sqrt(5.0) / 2.0
	t_3 = math.cos(x) - math.cos(y)
	t_4 = math.sin(y) - (math.sin(x) / 16.0)
	tmp = 0
	if x <= -0.0036:
		tmp = (2.0 + (t_3 * (t_4 * t_0))) / t_1
	elif x <= 6.5e+20:
		tmp = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * t_4) * (1.0 - math.cos(y)))) / t_1
	else:
		tmp = (2.0 + ((t_4 * t_3) * t_0)) / (3.0 * (1.0 + ((math.cos(x) * (t_2 - 0.5)) + (math.cos(y) * (1.5 - t_2)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * sin(x))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	t_2 = Float64(sqrt(5.0) / 2.0)
	t_3 = Float64(cos(x) - cos(y))
	t_4 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if (x <= -0.0036)
		tmp = Float64(Float64(2.0 + Float64(t_3 * Float64(t_4 * t_0))) / t_1);
	elseif (x <= 6.5e+20)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_4) * Float64(1.0 - cos(y)))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_4 * t_3) * t_0)) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_2 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * sin(x);
	t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	t_2 = sqrt(5.0) / 2.0;
	t_3 = cos(x) - cos(y);
	t_4 = sin(y) - (sin(x) / 16.0);
	tmp = 0.0;
	if (x <= -0.0036)
		tmp = (2.0 + (t_3 * (t_4 * t_0))) / t_1;
	elseif (x <= 6.5e+20)
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_4) * (1.0 - cos(y)))) / t_1;
	else
		tmp = (2.0 + ((t_4 * t_3) * t_0)) / (3.0 * (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0036], N[(N[(2.0 + N[(t$95$3 * N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 6.5e+20], 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] * t$95$4), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$4 * t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0035999999999999999

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 56.9%

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

   if -0.0035999999999999999 < x < 6.5e20

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 97.9%

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

   if 6.5e20 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}} \]

   4. Taylor expanded in y around 0 66.0%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0036:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 7: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.00375 \lor \neg \left(x \leq 0.032\right):\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_1\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)) (t_1 (- (sin y) (/ (sin x) 16.0))))
   (if (or (<= x -0.00375) (not (<= x 0.032)))
     (/
      (+ 2.0 (* (* t_1 (- (cos x) (cos y))) (* (sqrt 2.0) (sin x))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+
       2.0
       (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_1) (- 1.0 (cos y))))
      (*
       3.0
       (+
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))
        (+ 1.0 (* (+ (sqrt 5.0) -1.0) (+ 0.5 (* (* x x) -0.25))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if ((x <= -0.00375) || !(x <= 0.032)) {
		tmp = (2.0 + ((t_1 * (cos(x) - cos(y))) * (sqrt(2.0) * sin(x)))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_1) * (1.0 - cos(y)))) / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + ((sqrt(5.0) + -1.0) * (0.5 + ((x * x) * -0.25))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = sin(y) - (sin(x) / 16.0d0)
    if ((x <= (-0.00375d0)) .or. (.not. (x <= 0.032d0))) then
        tmp = (2.0d0 + ((t_1 * (cos(x) - cos(y))) * (sqrt(2.0d0) * sin(x)))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * t_1) * (1.0d0 - cos(y)))) / (3.0d0 * ((cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)) + (1.0d0 + ((sqrt(5.0d0) + (-1.0d0)) * (0.5d0 + ((x * x) * (-0.25d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = Math.sin(y) - (Math.sin(x) / 16.0);
	double tmp;
	if ((x <= -0.00375) || !(x <= 0.032)) {
		tmp = (2.0 + ((t_1 * (Math.cos(x) - Math.cos(y))) * (Math.sqrt(2.0) * Math.sin(x)))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * t_1) * (1.0 - Math.cos(y)))) / (3.0 * ((Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)) + (1.0 + ((Math.sqrt(5.0) + -1.0) * (0.5 + ((x * x) * -0.25))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = math.sin(y) - (math.sin(x) / 16.0)
	tmp = 0
	if (x <= -0.00375) or not (x <= 0.032):
		tmp = (2.0 + ((t_1 * (math.cos(x) - math.cos(y))) * (math.sqrt(2.0) * math.sin(x)))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * t_1) * (1.0 - math.cos(y)))) / (3.0 * ((math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)) + (1.0 + ((math.sqrt(5.0) + -1.0) * (0.5 + ((x * x) * -0.25))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if ((x <= -0.00375) || !(x <= 0.032))
		tmp = Float64(Float64(2.0 + Float64(Float64(t_1 * Float64(cos(x) - cos(y))) * Float64(sqrt(2.0) * sin(x)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_1) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) + Float64(1.0 + Float64(Float64(sqrt(5.0) + -1.0) * Float64(0.5 + Float64(Float64(x * x) * -0.25)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = sin(y) - (sin(x) / 16.0);
	tmp = 0.0;
	if ((x <= -0.00375) || ~((x <= 0.032)))
		tmp = (2.0 + ((t_1 * (cos(x) - cos(y))) * (sqrt(2.0) * sin(x)))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_1) * (1.0 - cos(y)))) / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + ((sqrt(5.0) + -1.0) * (0.5 + ((x * x) * -0.25))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00375], N[Not[LessEqual[x, 0.032]], $MachinePrecision]], N[(N[(2.0 + N[(N[(t$95$1 * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0037499999999999999 or 0.032000000000000001 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-lft-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-lft-in 99.0%

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

      5. associate-+l+ 99.0%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}} \]

   4. Taylor expanded in y around 0 61.0%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)} \]

   if -0.0037499999999999999 < x < 0.032000000000000001

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 98.8%

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

   3. Taylor expanded in x around 0 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(-0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(0.5 \cdot \left(\sqrt{5} - 1\right) + -0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\left(\sqrt{5} - 1\right) \cdot 0.5} + -0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\left(\sqrt{5} - 1\right) \cdot 0.5 + \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right) \cdot -0.25}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\left(\sqrt{5} - 1\right) \cdot 0.5 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left({x}^{2} \cdot -0.25\right)}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. distribute-lft-out 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-neg 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + \color{blue}{-1}\right) \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. unpow2 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00375 \lor \neg \left(x \leq 0.032\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)\right)\right)}\\ \end{array} \]

Alternative 8: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \sin x\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} + -1\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_5 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_3 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_2}{2}\right) + t_4\right)}\\ \mathbf{elif}\;x \leq 0.032:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_3\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(t_4 + \left(1 + t_2 \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_3 \cdot t_1\right) \cdot t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_5 - 0.5\right) + \cos y \cdot \left(1.5 - t_5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (sin x)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (+ (sqrt 5.0) -1.0))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_5 (/ (sqrt 5.0) 2.0)))
   (if (<= x -0.005)
     (/
      (+ 2.0 (* t_1 (* t_3 t_0)))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_2 2.0))) t_4)))
     (if (<= x 0.032)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_3)
          (- 1.0 (cos y))))
        (* 3.0 (+ t_4 (+ 1.0 (* t_2 (+ 0.5 (* (* x x) -0.25)))))))
       (/
        (+ 2.0 (* (* t_3 t_1) t_0))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_5 0.5)) (* (cos y) (- 1.5 t_5))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * sin(x);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) + -1.0;
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_5 = sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -0.005) {
		tmp = (2.0 + (t_1 * (t_3 * t_0))) / (3.0 * ((1.0 + (cos(x) * (t_2 / 2.0))) + t_4));
	} else if (x <= 0.032) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_3) * (1.0 - cos(y)))) / (3.0 * (t_4 + (1.0 + (t_2 * (0.5 + ((x * x) * -0.25))))));
	} else {
		tmp = (2.0 + ((t_3 * t_1) * t_0)) / (3.0 * (1.0 + ((cos(x) * (t_5 - 0.5)) + (cos(y) * (1.5 - t_5)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * sin(x)
    t_1 = cos(x) - cos(y)
    t_2 = sqrt(5.0d0) + (-1.0d0)
    t_3 = sin(y) - (sin(x) / 16.0d0)
    t_4 = cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)
    t_5 = sqrt(5.0d0) / 2.0d0
    if (x <= (-0.005d0)) then
        tmp = (2.0d0 + (t_1 * (t_3 * t_0))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_2 / 2.0d0))) + t_4))
    else if (x <= 0.032d0) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * t_3) * (1.0d0 - cos(y)))) / (3.0d0 * (t_4 + (1.0d0 + (t_2 * (0.5d0 + ((x * x) * (-0.25d0)))))))
    else
        tmp = (2.0d0 + ((t_3 * t_1) * t_0)) / (3.0d0 * (1.0d0 + ((cos(x) * (t_5 - 0.5d0)) + (cos(y) * (1.5d0 - t_5)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * Math.sin(x);
	double t_1 = Math.cos(x) - Math.cos(y);
	double t_2 = Math.sqrt(5.0) + -1.0;
	double t_3 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_4 = Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0);
	double t_5 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -0.005) {
		tmp = (2.0 + (t_1 * (t_3 * t_0))) / (3.0 * ((1.0 + (Math.cos(x) * (t_2 / 2.0))) + t_4));
	} else if (x <= 0.032) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * t_3) * (1.0 - Math.cos(y)))) / (3.0 * (t_4 + (1.0 + (t_2 * (0.5 + ((x * x) * -0.25))))));
	} else {
		tmp = (2.0 + ((t_3 * t_1) * t_0)) / (3.0 * (1.0 + ((Math.cos(x) * (t_5 - 0.5)) + (Math.cos(y) * (1.5 - t_5)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * math.sin(x)
	t_1 = math.cos(x) - math.cos(y)
	t_2 = math.sqrt(5.0) + -1.0
	t_3 = math.sin(y) - (math.sin(x) / 16.0)
	t_4 = math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)
	t_5 = math.sqrt(5.0) / 2.0
	tmp = 0
	if x <= -0.005:
		tmp = (2.0 + (t_1 * (t_3 * t_0))) / (3.0 * ((1.0 + (math.cos(x) * (t_2 / 2.0))) + t_4))
	elif x <= 0.032:
		tmp = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * t_3) * (1.0 - math.cos(y)))) / (3.0 * (t_4 + (1.0 + (t_2 * (0.5 + ((x * x) * -0.25))))))
	else:
		tmp = (2.0 + ((t_3 * t_1) * t_0)) / (3.0 * (1.0 + ((math.cos(x) * (t_5 - 0.5)) + (math.cos(y) * (1.5 - t_5)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * sin(x))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) + -1.0)
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_5 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if (x <= -0.005)
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(t_3 * t_0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0))) + t_4)));
	elseif (x <= 0.032)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_3) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(t_4 + Float64(1.0 + Float64(t_2 * Float64(0.5 + Float64(Float64(x * x) * -0.25)))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_3 * t_1) * t_0)) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_5 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * sin(x);
	t_1 = cos(x) - cos(y);
	t_2 = sqrt(5.0) + -1.0;
	t_3 = sin(y) - (sin(x) / 16.0);
	t_4 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	t_5 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if (x <= -0.005)
		tmp = (2.0 + (t_1 * (t_3 * t_0))) / (3.0 * ((1.0 + (cos(x) * (t_2 / 2.0))) + t_4));
	elseif (x <= 0.032)
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_3) * (1.0 - cos(y)))) / (3.0 * (t_4 + (1.0 + (t_2 * (0.5 + ((x * x) * -0.25))))));
	else
		tmp = (2.0 + ((t_3 * t_1) * t_0)) / (3.0 * (1.0 + ((cos(x) * (t_5 - 0.5)) + (cos(y) * (1.5 - t_5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.005], N[(N[(2.0 + N[(t$95$1 * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.032], 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] * t$95$3), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$4 + N[(1.0 + N[(t$95$2 * N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$3 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$5 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0050000000000000001

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 56.9%

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

   if -0.0050000000000000001 < x < 0.032000000000000001

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 98.8%

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

   3. Taylor expanded in x around 0 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(-0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(0.5 \cdot \left(\sqrt{5} - 1\right) + -0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\left(\sqrt{5} - 1\right) \cdot 0.5} + -0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\left(\sqrt{5} - 1\right) \cdot 0.5 + \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right) \cdot -0.25}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\left(\sqrt{5} - 1\right) \cdot 0.5 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left({x}^{2} \cdot -0.25\right)}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. distribute-lft-out 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-neg 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + \color{blue}{-1}\right) \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. unpow2 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.032000000000000001 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}} \]

   4. Taylor expanded in y around 0 64.7%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.032:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 9: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0085 \lor \neg \left(x \leq 0.032\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \sin x, t_0 \cdot \left(\cos x + -1\right), 2\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}} + 0.5 \cdot \left(\cos x \cdot t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot t_0\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + t_1 \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -0.0085) (not (<= x 0.032)))
     (/
      (fma (* (sqrt 2.0) (sin x)) (* t_0 (+ (cos x) -1.0)) 2.0)
      (*
       3.0
       (+
        1.0
        (+ (/ (cos y) (+ 1.5 (* 0.5 (sqrt 5.0)))) (* 0.5 (* (cos x) t_1))))))
     (/
      (+
       2.0
       (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_0) (- 1.0 (cos y))))
      (*
       3.0
       (+
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))
        (+ 1.0 (* t_1 (+ 0.5 (* (* x x) -0.25))))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.0085) || !(x <= 0.032)) {
		tmp = fma((sqrt(2.0) * sin(x)), (t_0 * (cos(x) + -1.0)), 2.0) / (3.0 * (1.0 + ((cos(y) / (1.5 + (0.5 * sqrt(5.0)))) + (0.5 * (cos(x) * t_1)))));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * t_0) * (1.0 - cos(y)))) / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + (t_1 * (0.5 + ((x * x) * -0.25))))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -0.0085) || !(x <= 0.032))
		tmp = Float64(fma(Float64(sqrt(2.0) * sin(x)), Float64(t_0 * Float64(cos(x) + -1.0)), 2.0) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(y) / Float64(1.5 + Float64(0.5 * sqrt(5.0)))) + Float64(0.5 * Float64(cos(x) * t_1))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * t_0) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) + Float64(1.0 + Float64(t_1 * Float64(0.5 + Float64(Float64(x * x) * -0.25)))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0085], N[Not[LessEqual[x, 0.032]], $MachinePrecision]], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[y], $MachinePrecision] / N[(1.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$1 * N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0085000000000000006 or 0.032000000000000001 < x

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      2. associate-*l* 99.0%

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

      3. fma-def 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. cos-neg 99.0%

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

      6. distribute-lft-in 99.0%

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

   3. Simplified 99.0%

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

      1. flip-- 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \color{blue}{\frac{1.5 \cdot 1.5 - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      2. metadata-eval 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      3. div-inv 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      4. metadata-eval 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      5. div-inv 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      6. metadata-eval 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      7. div-inv 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      8. metadata-eval 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   5. Applied egg-rr 98.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      2. rem-square-sqrt 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      3. cancel-sign-sub-inv 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      4. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      5. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      6. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      7. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{\color{blue}{1}}{1.5 + \sqrt{5} \cdot 0.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      8. +-commutative 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{1}{\color{blue}{\sqrt{5} \cdot 0.5 + 1.5}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      9. *-commutative 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{1}{\color{blue}{0.5 \cdot \sqrt{5}} + 1.5}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      10. fma-def 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   7. Simplified 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, \color{blue}{\frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   8. Taylor expanded in y around inf 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}}\right)\right)}} \]

   9. Taylor expanded in y around 0 61.1%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot \sin x}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}}\right)\right)} \]

   10. Taylor expanded in y around 0 57.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \sin x, \left(\sin y - \frac{\sin x}{16}\right) \cdot \color{blue}{\left(\cos x - 1\right)}, 2\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}}\right)\right)} \]

   if -0.0085000000000000006 < x < 0.032000000000000001

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 98.8%

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

   3. Taylor expanded in x around 0 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(-0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(0.5 \cdot \left(\sqrt{5} - 1\right) + -0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\left(\sqrt{5} - 1\right) \cdot 0.5} + -0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\left(\sqrt{5} - 1\right) \cdot 0.5 + \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right) \cdot -0.25}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\left(\sqrt{5} - 1\right) \cdot 0.5 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left({x}^{2} \cdot -0.25\right)}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. distribute-lft-out 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-neg 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + \color{blue}{-1}\right) \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. unpow2 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0085 \lor \neg \left(x \leq 0.032\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \sin x, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right), 2\right)}{3 \cdot \left(1 + \left(\frac{\cos y}{1.5 + 0.5 \cdot \sqrt{5}} + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)\right)\right)}\\ \end{array} \]

Alternative 10: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.00375 \lor \neg \left(x \leq 0.032\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(t_0 + \left(1 + t_1 \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -0.00375) (not (<= x 0.032)))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (pow (sin x) 2.0) (* (sqrt 2.0) -0.0625))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_1 2.0))) t_0)))
     (/
      (+
       2.0
       (*
        (*
         (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
         (- (sin y) (/ (sin x) 16.0)))
        (- 1.0 (cos y))))
      (* 3.0 (+ t_0 (+ 1.0 (* t_1 (+ 0.5 (* (* x x) -0.25))))))))))

C

double code(double x, double y) {
	double t_0 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.00375) || !(x <= 0.032)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(x), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + t_0));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (1.0 - cos(y)))) / (3.0 * (t_0 + (1.0 + (t_1 * (0.5 + ((x * x) * -0.25))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    if ((x <= (-0.00375d0)) .or. (.not. (x <= 0.032d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (-0.0625d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_1 / 2.0d0))) + t_0))
    else
        tmp = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (1.0d0 - cos(y)))) / (3.0d0 * (t_0 + (1.0d0 + (t_1 * (0.5d0 + ((x * x) * (-0.25d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0);
	double t_1 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.00375) || !(x <= 0.032)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (Math.cos(x) * (t_1 / 2.0))) + t_0));
	} else {
		tmp = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (1.0 - Math.cos(y)))) / (3.0 * (t_0 + (1.0 + (t_1 * (0.5 + ((x * x) * -0.25))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)
	t_1 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -0.00375) or not (x <= 0.032):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (math.cos(x) * (t_1 / 2.0))) + t_0))
	else:
		tmp = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (1.0 - math.cos(y)))) / (3.0 * (t_0 + (1.0 + (t_1 * (0.5 + ((x * x) * -0.25))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -0.00375) || !(x <= 0.032))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + t_0)));
	else
		tmp = 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(1.0 - cos(y)))) / Float64(3.0 * Float64(t_0 + Float64(1.0 + Float64(t_1 * Float64(0.5 + Float64(Float64(x * x) * -0.25)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	t_1 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -0.00375) || ~((x <= 0.032)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(x) ^ 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + t_0));
	else
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (1.0 - cos(y)))) / (3.0 * (t_0 + (1.0 + (t_1 * (0.5 + ((x * x) * -0.25))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.00375], N[Not[LessEqual[x, 0.032]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(1.0 + N[(t$95$1 * N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0037499999999999999 or 0.032000000000000001 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 57.5%

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

      1. associate-*r* 57.5%

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

      2. *-commutative 57.5%

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

   4. Simplified 57.5%

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

   if -0.0037499999999999999 < x < 0.032000000000000001

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 98.8%

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

   3. Taylor expanded in x around 0 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(-0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(0.5 \cdot \left(\sqrt{5} - 1\right) + -0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\left(\sqrt{5} - 1\right) \cdot 0.5} + -0.25 \cdot \left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\left(\sqrt{5} - 1\right) \cdot 0.5 + \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right) \cdot -0.25}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\left(\sqrt{5} - 1\right) \cdot 0.5 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left({x}^{2} \cdot -0.25\right)}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. distribute-lft-out 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-neg 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + \color{blue}{-1}\right) \cdot \left(0.5 + {x}^{2} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. unpow2 98.8%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.25\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00375 \lor \neg \left(x \leq 0.032\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.25\right)\right)\right)}\\ \end{array} \]

Alternative 11: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.003 \lor \neg \left(x \leq 6.5 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.003) (not (<= x 6.5e+20)))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (pow (sin x) 2.0) (* (sqrt 2.0) -0.0625))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
        (* (sin y) (- 1.0 (cos y)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.003) || !(x <= 6.5e+20)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(x), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) * (1.0 - cos(y))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-0.003d0)) .or. (.not. (x <= 6.5d+20))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (-0.0625d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) * (1.0d0 - cos(y))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.003) || !(x <= 6.5e+20)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) * (1.0 - Math.cos(y))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -0.003) or not (x <= 6.5e+20):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) * (1.0 - math.cos(y))))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.003) || !(x <= 6.5e+20))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) * Float64(1.0 - cos(y))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -0.003) || ~((x <= 6.5e+20)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(x) ^ 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) * (1.0 - cos(y))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.003], N[Not[LessEqual[x, 6.5e+20]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + 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[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0030000000000000001 or 6.5e20 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 58.1%

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

      1. associate-*r* 58.1%

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

      2. *-commutative 58.1%

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

   4. Simplified 58.1%

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

   if -0.0030000000000000001 < x < 6.5e20

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. distribute-lft-in 99.8%

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

      3. cos-neg 99.8%

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

      4. distribute-lft-in 99.7%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}} \]

   4. Taylor expanded in x around 0 97.4%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.003 \lor \neg \left(x \leq 6.5 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 12: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-5} \lor \neg \left(y \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_0 \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + t_0 \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot t_1\right) - 0.5 \cdot \sqrt{5}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) -0.0625)) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= y -5.8e-5) (not (<= y 1.05e-8)))
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* t_0 (pow (sin y) 2.0))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_1 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (*
       0.3333333333333333
       (+ 2.0 (* t_0 (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
      (+ 2.5 (- (* 0.5 (* (cos x) t_1)) (* 0.5 (sqrt 5.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * -0.0625;
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -5.8e-5) || !(y <= 1.05e-8)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_0 * pow(sin(y), 2.0)))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / (2.5 + ((0.5 * (cos(x) * t_1)) - (0.5 * sqrt(5.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (-0.0625d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    if ((y <= (-5.8d-5)) .or. (.not. (y <= 1.05d-8))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * (t_0 * (sin(y) ** 2.0d0)))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_1 / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (0.3333333333333333d0 * (2.0d0 + (t_0 * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / (2.5d0 + ((0.5d0 * (cos(x) * t_1)) - (0.5d0 * sqrt(5.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * -0.0625;
	double t_1 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -5.8e-5) || !(y <= 1.05e-8)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (t_0 * Math.pow(Math.sin(y), 2.0)))) / (3.0 * ((1.0 + (Math.cos(x) * (t_1 / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / (2.5 + ((0.5 * (Math.cos(x) * t_1)) - (0.5 * Math.sqrt(5.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * -0.0625
	t_1 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (y <= -5.8e-5) or not (y <= 1.05e-8):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (t_0 * math.pow(math.sin(y), 2.0)))) / (3.0 * ((1.0 + (math.cos(x) * (t_1 / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / (2.5 + ((0.5 * (math.cos(x) * t_1)) - (0.5 * math.sqrt(5.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * -0.0625)
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((y <= -5.8e-5) || !(y <= 1.05e-8))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_0 * (sin(y) ^ 2.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(t_0 * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / Float64(2.5 + Float64(Float64(0.5 * Float64(cos(x) * t_1)) - Float64(0.5 * sqrt(5.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * -0.0625;
	t_1 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((y <= -5.8e-5) || ~((y <= 1.05e-8)))
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_0 * (sin(y) ^ 2.0)))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / (2.5 + ((0.5 * (cos(x) * t_1)) - (0.5 * sqrt(5.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[y, -5.8e-5], N[Not[LessEqual[y, 1.05e-8]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(2.0 + N[(t$95$0 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.8e-5 or 1.04999999999999997e-8 < y

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 62.1%

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

      1. associate-*r* 59.8%

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

   4. Simplified 62.1%

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

   if -5.8e-5 < y < 1.04999999999999997e-8

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. fma-def 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. cos-neg 99.6%

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

      6. distribute-lft-in 99.6%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)}} \]

   4. Taylor expanded in y around 0 99.7%

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

      1. associate-*r* 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      3. *-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x - 1\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      4. sub-neg 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + \color{blue}{-1}\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + -1\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   7. Taylor expanded in y around 0 99.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}}} \]

      2. associate-*r* 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \color{blue}{\left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      3. sub-neg 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      5. associate--l+ 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\color{blue}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)}} \]

      6. sub-neg 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)} \]

      7. metadata-eval 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} + \color{blue}{-1}\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)} \]

   9. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} + -1\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-5} \lor \neg \left(y \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) - 0.5 \cdot \sqrt{5}\right)}\\ \end{array} \]

Alternative 13: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{5}\\ \mathbf{if}\;x \leq -0.000115 \lor \neg \left(x \leq 3.5 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt 5.0))))
   (if (or (<= x -0.000115) (not (<= x 3.5e-13)))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (pow (sin x) 2.0) (* (sqrt 2.0) -0.0625))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
        (* (sin y) (- 1.0 (cos y)))))
      (* 3.0 (+ 1.0 (- (+ t_0 (* (cos y) (- 1.5 t_0))) 0.5)))))))

C

double code(double x, double y) {
	double t_0 = 0.5 * sqrt(5.0);
	double tmp;
	if ((x <= -0.000115) || !(x <= 3.5e-13)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(x), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) * (1.0 - cos(y))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt(5.0d0)
    if ((x <= (-0.000115d0)) .or. (.not. (x <= 3.5d-13))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (-0.0625d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) * (1.0d0 - cos(y))))) / (3.0d0 * (1.0d0 + ((t_0 + (cos(y) * (1.5d0 - t_0))) - 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * Math.sqrt(5.0);
	double tmp;
	if ((x <= -0.000115) || !(x <= 3.5e-13)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) * (1.0 - Math.cos(y))))) / (3.0 * (1.0 + ((t_0 + (Math.cos(y) * (1.5 - t_0))) - 0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * math.sqrt(5.0)
	tmp = 0
	if (x <= -0.000115) or not (x <= 3.5e-13):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) * (1.0 - math.cos(y))))) / (3.0 * (1.0 + ((t_0 + (math.cos(y) * (1.5 - t_0))) - 0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.000115) || !(x <= 3.5e-13))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) * Float64(1.0 - cos(y))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * sqrt(5.0);
	tmp = 0.0;
	if ((x <= -0.000115) || ~((x <= 3.5e-13)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(x) ^ 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) * (1.0 - cos(y))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.000115], N[Not[LessEqual[x, 3.5e-13]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + 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[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e-4 or 3.5000000000000002e-13 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 58.2%

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

      1. associate-*r* 58.2%

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

      2. *-commutative 58.2%

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

   4. Simplified 58.2%

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

   if -1.15e-4 < x < 3.5000000000000002e-13

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. distribute-lft-in 99.8%

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

      3. cos-neg 99.8%

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

      4. distribute-lft-in 99.7%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}} \]

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)} \]

   5. Taylor expanded in x around 0 99.0%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right)}{3 \cdot \left(1 + \color{blue}{\left(\left(0.5 \cdot \sqrt{5} + \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right) - 0.5\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000115 \lor \neg \left(x \leq 3.5 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 14: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sqrt{2} \cdot -0.0625\\ t_2 := 2 + \left(\cos x - \cos y\right) \cdot \left(t_1 \cdot {\sin y}^{2}\right)\\ t_3 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + t_3\right)}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + t_1 \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot t_0\right) - 0.5 \cdot \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(t_3 + \left(1 + \frac{t_0}{\frac{2}{\cos x}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (* (sqrt 2.0) -0.0625))
        (t_2 (+ 2.0 (* (- (cos x) (cos y)) (* t_1 (pow (sin y) 2.0)))))
        (t_3 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))
   (if (<= y -1.25e-6)
     (/ t_2 (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) t_3)))
     (if (<= y 1.05e-8)
       (/
        (*
         0.3333333333333333
         (+ 2.0 (* t_1 (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
        (+ 2.5 (- (* 0.5 (* (cos x) t_0)) (* 0.5 (sqrt 5.0)))))
       (/ t_2 (* 3.0 (+ t_3 (+ 1.0 (/ t_0 (/ 2.0 (cos x)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = sqrt(2.0) * -0.0625;
	double t_2 = 2.0 + ((cos(x) - cos(y)) * (t_1 * pow(sin(y), 2.0)));
	double t_3 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double tmp;
	if (y <= -1.25e-6) {
		tmp = t_2 / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + t_3));
	} else if (y <= 1.05e-8) {
		tmp = (0.3333333333333333 * (2.0 + (t_1 * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / (2.5 + ((0.5 * (cos(x) * t_0)) - (0.5 * sqrt(5.0))));
	} else {
		tmp = t_2 / (3.0 * (t_3 + (1.0 + (t_0 / (2.0 / cos(x))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    t_1 = sqrt(2.0d0) * (-0.0625d0)
    t_2 = 2.0d0 + ((cos(x) - cos(y)) * (t_1 * (sin(y) ** 2.0d0)))
    t_3 = cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)
    if (y <= (-1.25d-6)) then
        tmp = t_2 / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 / 2.0d0))) + t_3))
    else if (y <= 1.05d-8) then
        tmp = (0.3333333333333333d0 * (2.0d0 + (t_1 * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / (2.5d0 + ((0.5d0 * (cos(x) * t_0)) - (0.5d0 * sqrt(5.0d0))))
    else
        tmp = t_2 / (3.0d0 * (t_3 + (1.0d0 + (t_0 / (2.0d0 / cos(x))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double t_1 = Math.sqrt(2.0) * -0.0625;
	double t_2 = 2.0 + ((Math.cos(x) - Math.cos(y)) * (t_1 * Math.pow(Math.sin(y), 2.0)));
	double t_3 = Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0);
	double tmp;
	if (y <= -1.25e-6) {
		tmp = t_2 / (3.0 * ((1.0 + (Math.cos(x) * (t_0 / 2.0))) + t_3));
	} else if (y <= 1.05e-8) {
		tmp = (0.3333333333333333 * (2.0 + (t_1 * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / (2.5 + ((0.5 * (Math.cos(x) * t_0)) - (0.5 * Math.sqrt(5.0))));
	} else {
		tmp = t_2 / (3.0 * (t_3 + (1.0 + (t_0 / (2.0 / Math.cos(x))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	t_1 = math.sqrt(2.0) * -0.0625
	t_2 = 2.0 + ((math.cos(x) - math.cos(y)) * (t_1 * math.pow(math.sin(y), 2.0)))
	t_3 = math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)
	tmp = 0
	if y <= -1.25e-6:
		tmp = t_2 / (3.0 * ((1.0 + (math.cos(x) * (t_0 / 2.0))) + t_3))
	elif y <= 1.05e-8:
		tmp = (0.3333333333333333 * (2.0 + (t_1 * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / (2.5 + ((0.5 * (math.cos(x) * t_0)) - (0.5 * math.sqrt(5.0))))
	else:
		tmp = t_2 / (3.0 * (t_3 + (1.0 + (t_0 / (2.0 / math.cos(x))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(sqrt(2.0) * -0.0625)
	t_2 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_1 * (sin(y) ^ 2.0))))
	t_3 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	tmp = 0.0
	if (y <= -1.25e-6)
		tmp = Float64(t_2 / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + t_3)));
	elseif (y <= 1.05e-8)
		tmp = Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(t_1 * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / Float64(2.5 + Float64(Float64(0.5 * Float64(cos(x) * t_0)) - Float64(0.5 * sqrt(5.0)))));
	else
		tmp = Float64(t_2 / Float64(3.0 * Float64(t_3 + Float64(1.0 + Float64(t_0 / Float64(2.0 / cos(x)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	t_1 = sqrt(2.0) * -0.0625;
	t_2 = 2.0 + ((cos(x) - cos(y)) * (t_1 * (sin(y) ^ 2.0)));
	t_3 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	tmp = 0.0;
	if (y <= -1.25e-6)
		tmp = t_2 / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + t_3));
	elseif (y <= 1.05e-8)
		tmp = (0.3333333333333333 * (2.0 + (t_1 * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / (2.5 + ((0.5 * (cos(x) * t_0)) - (0.5 * sqrt(5.0))));
	else
		tmp = t_2 / (3.0 * (t_3 + (1.0 + (t_0 / (2.0 / cos(x))))));
	end
	tmp_2 = 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[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.25e-6], N[(t$95$2 / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e-8], N[(N[(0.3333333333333333 * N[(2.0 + N[(t$95$1 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 * N[(t$95$3 + N[(1.0 + N[(t$95$0 / N[(2.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2500000000000001e-6

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 58.5%

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

      1. associate-*r* 56.9%

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

   4. Simplified 58.5%

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

   if -1.2500000000000001e-6 < y < 1.04999999999999997e-8

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. fma-def 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. cos-neg 99.6%

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

      6. distribute-lft-in 99.6%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)}} \]

   4. Taylor expanded in y around 0 99.7%

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

      1. associate-*r* 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      3. *-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x - 1\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      4. sub-neg 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + \color{blue}{-1}\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + -1\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   7. Taylor expanded in y around 0 99.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}}} \]

      2. associate-*r* 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \color{blue}{\left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      3. sub-neg 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      5. associate--l+ 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\color{blue}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)}} \]

      6. sub-neg 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)} \]

      7. metadata-eval 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} + \color{blue}{-1}\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)} \]

   9. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} + -1\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   if 1.04999999999999997e-8 < y

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 65.8%

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

      1. associate-*r* 62.8%

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

   4. Simplified 65.8%

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

      1. associate-*l/ 65.8%

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

      2. sub-neg 65.8%

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

      3. metadata-eval 65.8%

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

   6. Applied egg-rr 65.8%

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

      1. metadata-eval 65.8%

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

      2. sub-neg 65.8%

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

      3. associate-/l* 65.8%

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

      4. sub-neg 65.8%

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

      5. metadata-eval 65.8%

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

   8. Simplified 65.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) - 0.5 \cdot \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\right)\right)}\\ \end{array} \]

Alternative 15: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{-6} \lor \neg \left(y \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(t_0 \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + t_0 \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot t_1\right) - 0.5 \cdot \sqrt{5}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) -0.0625)) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= y -1.6e-6) (not (<= y 1.05e-8)))
     (/
      (+ 2.0 (* (- 1.0 (cos y)) (* t_0 (pow (sin y) 2.0))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_1 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (*
       0.3333333333333333
       (+ 2.0 (* t_0 (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
      (+ 2.5 (- (* 0.5 (* (cos x) t_1)) (* 0.5 (sqrt 5.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * -0.0625;
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -1.6e-6) || !(y <= 1.05e-8)) {
		tmp = (2.0 + ((1.0 - cos(y)) * (t_0 * pow(sin(y), 2.0)))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / (2.5 + ((0.5 * (cos(x) * t_1)) - (0.5 * sqrt(5.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (-0.0625d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    if ((y <= (-1.6d-6)) .or. (.not. (y <= 1.05d-8))) then
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (t_0 * (sin(y) ** 2.0d0)))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_1 / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (0.3333333333333333d0 * (2.0d0 + (t_0 * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / (2.5d0 + ((0.5d0 * (cos(x) * t_1)) - (0.5d0 * sqrt(5.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * -0.0625;
	double t_1 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -1.6e-6) || !(y <= 1.05e-8)) {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (t_0 * Math.pow(Math.sin(y), 2.0)))) / (3.0 * ((1.0 + (Math.cos(x) * (t_1 / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / (2.5 + ((0.5 * (Math.cos(x) * t_1)) - (0.5 * Math.sqrt(5.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * -0.0625
	t_1 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (y <= -1.6e-6) or not (y <= 1.05e-8):
		tmp = (2.0 + ((1.0 - math.cos(y)) * (t_0 * math.pow(math.sin(y), 2.0)))) / (3.0 * ((1.0 + (math.cos(x) * (t_1 / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / (2.5 + ((0.5 * (math.cos(x) * t_1)) - (0.5 * math.sqrt(5.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * -0.0625)
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((y <= -1.6e-6) || !(y <= 1.05e-8))
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(t_0 * (sin(y) ^ 2.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(t_0 * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / Float64(2.5 + Float64(Float64(0.5 * Float64(cos(x) * t_1)) - Float64(0.5 * sqrt(5.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * -0.0625;
	t_1 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((y <= -1.6e-6) || ~((y <= 1.05e-8)))
		tmp = (2.0 + ((1.0 - cos(y)) * (t_0 * (sin(y) ^ 2.0)))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / (2.5 + ((0.5 * (cos(x) * t_1)) - (0.5 * sqrt(5.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[y, -1.6e-6], N[Not[LessEqual[y, 1.05e-8]], $MachinePrecision]], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(2.0 + N[(t$95$0 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5999999999999999e-6 or 1.04999999999999997e-8 < y

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 62.1%

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

      1. associate-*r* 59.8%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in x around 0 61.9%

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

   if -1.5999999999999999e-6 < y < 1.04999999999999997e-8

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. fma-def 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. cos-neg 99.6%

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

      6. distribute-lft-in 99.6%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)}} \]

   4. Taylor expanded in y around 0 99.7%

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

      1. associate-*r* 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      3. *-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x - 1\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      4. sub-neg 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + \color{blue}{-1}\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + -1\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   7. Taylor expanded in y around 0 99.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}}} \]

      2. associate-*r* 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \color{blue}{\left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      3. sub-neg 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      5. associate--l+ 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\color{blue}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)}} \]

      6. sub-neg 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)} \]

      7. metadata-eval 99.7%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} + \color{blue}{-1}\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)} \]

   9. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} + -1\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-6} \lor \neg \left(y \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) - 0.5 \cdot \sqrt{5}\right)}\\ \end{array} \]

Alternative 16: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.000115 \lor \neg \left(x \leq 3.5 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + t_0 \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot t_1\right) - 0.5 \cdot \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(t_0 \cdot {\sin y}^{2}\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + 0.5 \cdot t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) -0.0625)) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -0.000115) (not (<= x 3.5e-13)))
     (/
      (*
       0.3333333333333333
       (+ 2.0 (* t_0 (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
      (+ 2.5 (- (* 0.5 (* (cos x) t_1)) (* 0.5 (sqrt 5.0)))))
     (/
      (+ 2.0 (* (- 1.0 (cos y)) (* t_0 (pow (sin y) 2.0))))
      (*
       3.0
       (+ (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)) (+ 1.0 (* 0.5 t_1))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * -0.0625;
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.000115) || !(x <= 3.5e-13)) {
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / (2.5 + ((0.5 * (cos(x) * t_1)) - (0.5 * sqrt(5.0))));
	} else {
		tmp = (2.0 + ((1.0 - cos(y)) * (t_0 * pow(sin(y), 2.0)))) / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + (0.5 * t_1))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (-0.0625d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    if ((x <= (-0.000115d0)) .or. (.not. (x <= 3.5d-13))) then
        tmp = (0.3333333333333333d0 * (2.0d0 + (t_0 * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / (2.5d0 + ((0.5d0 * (cos(x) * t_1)) - (0.5d0 * sqrt(5.0d0))))
    else
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (t_0 * (sin(y) ** 2.0d0)))) / (3.0d0 * ((cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)) + (1.0d0 + (0.5d0 * t_1))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * -0.0625;
	double t_1 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.000115) || !(x <= 3.5e-13)) {
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / (2.5 + ((0.5 * (Math.cos(x) * t_1)) - (0.5 * Math.sqrt(5.0))));
	} else {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (t_0 * Math.pow(Math.sin(y), 2.0)))) / (3.0 * ((Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)) + (1.0 + (0.5 * t_1))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * -0.0625
	t_1 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -0.000115) or not (x <= 3.5e-13):
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / (2.5 + ((0.5 * (math.cos(x) * t_1)) - (0.5 * math.sqrt(5.0))))
	else:
		tmp = (2.0 + ((1.0 - math.cos(y)) * (t_0 * math.pow(math.sin(y), 2.0)))) / (3.0 * ((math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)) + (1.0 + (0.5 * t_1))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * -0.0625)
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -0.000115) || !(x <= 3.5e-13))
		tmp = Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(t_0 * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / Float64(2.5 + Float64(Float64(0.5 * Float64(cos(x) * t_1)) - Float64(0.5 * sqrt(5.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(t_0 * (sin(y) ^ 2.0)))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) + Float64(1.0 + Float64(0.5 * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * -0.0625;
	t_1 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -0.000115) || ~((x <= 3.5e-13)))
		tmp = (0.3333333333333333 * (2.0 + (t_0 * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / (2.5 + ((0.5 * (cos(x) * t_1)) - (0.5 * sqrt(5.0))));
	else
		tmp = (2.0 + ((1.0 - cos(y)) * (t_0 * (sin(y) ^ 2.0)))) / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + (0.5 * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.000115], N[Not[LessEqual[x, 3.5e-13]], $MachinePrecision]], N[(N[(0.3333333333333333 * N[(2.0 + N[(t$95$0 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e-4 or 3.5000000000000002e-13 < x

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      2. associate-*l* 99.0%

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

      3. fma-def 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. cos-neg 99.0%

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

      6. distribute-lft-in 99.0%

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

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)}} \]

   4. Taylor expanded in y around 0 58.2%

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

      1. associate-*r* 58.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      2. metadata-eval 58.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      3. *-commutative 58.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x - 1\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      4. sub-neg 58.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      5. metadata-eval 58.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + \color{blue}{-1}\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

      6. metadata-eval 58.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + -1\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. Simplified 58.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   7. Taylor expanded in y around 0 56.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 56.5%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}}} \]

      2. associate-*r* 56.5%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \color{blue}{\left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      3. sub-neg 56.5%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      4. metadata-eval 56.5%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

      5. associate--l+ 56.6%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\color{blue}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)}} \]

      6. sub-neg 56.6%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)} \]

      7. metadata-eval 56.6%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} + \color{blue}{-1}\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)} \]

   9. Simplified 56.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} + -1\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   if -1.15e-4 < x < 3.5000000000000002e-13

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.4%

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

   3. Taylor expanded in x around 0 99.3%

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

   4. Taylor expanded in x around 0 98.8%

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

      1. associate-*r* 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000115 \lor \neg \left(x \leq 3.5 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) - 0.5 \cdot \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + 0.5 \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \end{array} \]

Alternative 17: 60.2% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
   (- (+ 2.5 (* 0.5 (* (cos x) (+ (sqrt 5.0) -1.0)))) (* 0.5 (sqrt 5.0))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / ((2.5 + (0.5 * (cos(x) * (sqrt(5.0) + -1.0)))) - (0.5 * sqrt(5.0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / ((2.5d0 + (0.5d0 * (cos(x) * (sqrt(5.0d0) + (-1.0d0))))) - (0.5d0 * sqrt(5.0d0))))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / ((2.5 + (0.5 * (Math.cos(x) * (Math.sqrt(5.0) + -1.0)))) - (0.5 * Math.sqrt(5.0))));
}

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / ((2.5 + (0.5 * (math.cos(x) * (math.sqrt(5.0) + -1.0)))) - (0.5 * math.sqrt(5.0))))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / Float64(Float64(2.5 + Float64(0.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))) - Float64(0.5 * sqrt(5.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / ((2.5 + (0.5 * (cos(x) * (sqrt(5.0) + -1.0)))) - (0.5 * sqrt(5.0))));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.5 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. cos-neg 99.4%

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

   6. distribute-lft-in 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)}} \]

4. Taylor expanded in y around 0 62.0%

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

   1. associate-*r* 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   2. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   3. *-commutative 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x - 1\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   4. sub-neg 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   5. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + \color{blue}{-1}\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + -1\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

6. Simplified 62.0%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

7. Taylor expanded in y around 0 59.8%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}}} \]

8. Final simplification 59.8%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

Alternative 18: 60.3% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (*
   0.3333333333333333
   (+ 2.0 (* (* (sqrt 2.0) -0.0625) (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
  (+ 2.5 (- (* 0.5 (* (cos x) (+ (sqrt 5.0) -1.0))) (* 0.5 (sqrt 5.0))))))

C

double code(double x, double y) {
	return (0.3333333333333333 * (2.0 + ((sqrt(2.0) * -0.0625) * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / (2.5 + ((0.5 * (cos(x) * (sqrt(5.0) + -1.0))) - (0.5 * sqrt(5.0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.3333333333333333d0 * (2.0d0 + ((sqrt(2.0d0) * (-0.0625d0)) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / (2.5d0 + ((0.5d0 * (cos(x) * (sqrt(5.0d0) + (-1.0d0)))) - (0.5d0 * sqrt(5.0d0))))
end function

Java

public static double code(double x, double y) {
	return (0.3333333333333333 * (2.0 + ((Math.sqrt(2.0) * -0.0625) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / (2.5 + ((0.5 * (Math.cos(x) * (Math.sqrt(5.0) + -1.0))) - (0.5 * Math.sqrt(5.0))));
}

Python

def code(x, y):
	return (0.3333333333333333 * (2.0 + ((math.sqrt(2.0) * -0.0625) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / (2.5 + ((0.5 * (math.cos(x) * (math.sqrt(5.0) + -1.0))) - (0.5 * math.sqrt(5.0))))

Julia

function code(x, y)
	return Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(Float64(sqrt(2.0) * -0.0625) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / Float64(2.5 + Float64(Float64(0.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) - Float64(0.5 * sqrt(5.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = (0.3333333333333333 * (2.0 + ((sqrt(2.0) * -0.0625) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / (2.5 + ((0.5 * (cos(x) * (sqrt(5.0) + -1.0))) - (0.5 * sqrt(5.0))));
end

Wolfram

code[x_, y_] := N[(N[(0.3333333333333333 * N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. cos-neg 99.4%

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

   6. distribute-lft-in 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)}} \]

4. Taylor expanded in y around 0 62.0%

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

   1. associate-*r* 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   2. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   3. *-commutative 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x - 1\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   4. sub-neg 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   5. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + \color{blue}{-1}\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + -1\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

6. Simplified 62.0%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

7. Taylor expanded in y around 0 59.8%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}}} \]
8. Step-by-step derivation

   1. associate-*r/ 59.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}}} \]

   2. associate-*r* 59.8%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \color{blue}{\left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

   3. sub-neg 59.8%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

   4. metadata-eval 59.8%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)\right) - 0.5 \cdot \sqrt{5}} \]

   5. associate--l+ 59.8%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{\color{blue}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   6. sub-neg 59.8%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)} \]

   7. metadata-eval 59.8%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} + \color{blue}{-1}\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)} \]

9. Simplified 59.8%

   \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + \left(-0.0625 \cdot \sqrt{2}\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(0.5 \cdot \left(\left(\sqrt{5} + -1\right) \cdot \cos x\right) - 0.5 \cdot \sqrt{5}\right)}} \]

10. Final simplification 59.8%

    \[\leadsto \frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) - 0.5 \cdot \sqrt{5}\right)} \]

Alternative 19: 42.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{\frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (log
  (exp
   (/
    0.6666666666666666
    (+
     1.0
     (fma 0.5 (+ (sqrt 5.0) -1.0) (* (cos y) (+ 1.5 (* (sqrt 5.0) -0.5)))))))))

C

double code(double x, double y) {
	return log(exp((0.6666666666666666 / (1.0 + fma(0.5, (sqrt(5.0) + -1.0), (cos(y) * (1.5 + (sqrt(5.0) * -0.5))))))));
}

Julia

function code(x, y)
	return log(exp(Float64(0.6666666666666666 / Float64(1.0 + fma(0.5, Float64(sqrt(5.0) + -1.0), Float64(cos(y) * Float64(1.5 + Float64(sqrt(5.0) * -0.5))))))))
end

Wolfram

code[x_, y_] := N[Log[N[Exp[N[(0.6666666666666666 / N[(1.0 + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. cos-neg 99.4%

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

   6. distribute-lft-in 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)}} \]

4. Taylor expanded in y around 0 62.0%

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

   1. associate-*r* 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   2. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   3. *-commutative 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x - 1\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   4. sub-neg 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   5. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + \color{blue}{-1}\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + -1\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

6. Simplified 62.0%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

7. Taylor expanded in x around 0 45.6%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\sqrt{5} - 1\right) + \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}} \]
8. Step-by-step derivation

   1. add-log-exp 45.7%

      \[\leadsto \color{blue}{\log \left(e^{\frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\sqrt{5} - 1\right) + \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}}\right)} \]

   2. fma-def 45.7%

      \[\leadsto \log \left(e^{\frac{0.6666666666666666}{1 + \color{blue}{\mathsf{fma}\left(0.5, \sqrt{5} - 1, \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}}}\right) \]

   3. sub-neg 45.7%

      \[\leadsto \log \left(e^{\frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5} + \left(-1\right)}, \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}}\right) \]

   4. metadata-eval 45.7%

      \[\leadsto \log \left(e^{\frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, \sqrt{5} + \color{blue}{-1}, \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}}\right) \]

   5. *-commutative 45.7%

      \[\leadsto \log \left(e^{\frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \color{blue}{\cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)}\right)}}\right) \]

   6. cancel-sign-sub-inv 45.7%

      \[\leadsto \log \left(e^{\frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \cos y \cdot \color{blue}{\left(1.5 + \left(-0.5\right) \cdot \sqrt{5}\right)}\right)}}\right) \]

   7. metadata-eval 45.7%

      \[\leadsto \log \left(e^{\frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \cos y \cdot \left(1.5 + \color{blue}{-0.5} \cdot \sqrt{5}\right)\right)}}\right) \]

9. Applied egg-rr 45.7%

   \[\leadsto \color{blue}{\log \left(e^{\frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \cos y \cdot \left(1.5 + -0.5 \cdot \sqrt{5}\right)\right)}}\right)} \]

10. Final simplification 45.7%

    \[\leadsto \log \left(e^{\frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}}\right) \]

Alternative 20: 42.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}{0.6666666666666666}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (pow
  (/
   (+
    1.0
    (fma 0.5 (+ (sqrt 5.0) -1.0) (* (cos y) (+ 1.5 (* (sqrt 5.0) -0.5)))))
   0.6666666666666666)
  -1.0))

C

double code(double x, double y) {
	return pow(((1.0 + fma(0.5, (sqrt(5.0) + -1.0), (cos(y) * (1.5 + (sqrt(5.0) * -0.5))))) / 0.6666666666666666), -1.0);
}

Julia

function code(x, y)
	return Float64(Float64(1.0 + fma(0.5, Float64(sqrt(5.0) + -1.0), Float64(cos(y) * Float64(1.5 + Float64(sqrt(5.0) * -0.5))))) / 0.6666666666666666) ^ -1.0
end

Wolfram

code[x_, y_] := N[Power[N[(N[(1.0 + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. cos-neg 99.4%

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

   6. distribute-lft-in 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)}} \]

4. Taylor expanded in y around 0 62.0%

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

   1. associate-*r* 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   2. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   3. *-commutative 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x - 1\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   4. sub-neg 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   5. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + \color{blue}{-1}\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + -1\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

6. Simplified 62.0%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

7. Taylor expanded in x around 0 45.6%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\sqrt{5} - 1\right) + \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}} \]
8. Step-by-step derivation

   1. clear-num 45.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + \left(0.5 \cdot \left(\sqrt{5} - 1\right) + \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}{0.6666666666666666}}} \]

   2. inv-pow 45.7%

      \[\leadsto \color{blue}{{\left(\frac{1 + \left(0.5 \cdot \left(\sqrt{5} - 1\right) + \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}{0.6666666666666666}\right)}^{-1}} \]

   3. fma-def 45.7%

      \[\leadsto {\left(\frac{1 + \color{blue}{\mathsf{fma}\left(0.5, \sqrt{5} - 1, \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}}{0.6666666666666666}\right)}^{-1} \]

   4. sub-neg 45.7%

      \[\leadsto {\left(\frac{1 + \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5} + \left(-1\right)}, \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}{0.6666666666666666}\right)}^{-1} \]

   5. metadata-eval 45.7%

      \[\leadsto {\left(\frac{1 + \mathsf{fma}\left(0.5, \sqrt{5} + \color{blue}{-1}, \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}{0.6666666666666666}\right)}^{-1} \]

   6. *-commutative 45.7%

      \[\leadsto {\left(\frac{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \color{blue}{\cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)}\right)}{0.6666666666666666}\right)}^{-1} \]

   7. cancel-sign-sub-inv 45.7%

      \[\leadsto {\left(\frac{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \cos y \cdot \color{blue}{\left(1.5 + \left(-0.5\right) \cdot \sqrt{5}\right)}\right)}{0.6666666666666666}\right)}^{-1} \]

   8. metadata-eval 45.7%

      \[\leadsto {\left(\frac{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \cos y \cdot \left(1.5 + \color{blue}{-0.5} \cdot \sqrt{5}\right)\right)}{0.6666666666666666}\right)}^{-1} \]

9. Applied egg-rr 45.7%

   \[\leadsto \color{blue}{{\left(\frac{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \cos y \cdot \left(1.5 + -0.5 \cdot \sqrt{5}\right)\right)}{0.6666666666666666}\right)}^{-1}} \]

10. Final simplification 45.7%

    \[\leadsto {\left(\frac{1 + \mathsf{fma}\left(0.5, \sqrt{5} + -1, \cos y \cdot \left(1.5 + \sqrt{5} \cdot -0.5\right)\right)}{0.6666666666666666}\right)}^{-1} \]

Alternative 21: 42.2% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (+
   1.0
   (fma (+ 1.5 (* (sqrt 5.0) -0.5)) (cos y) (* 0.5 (+ (sqrt 5.0) -1.0))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. cos-neg 99.4%

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

   6. distribute-lft-in 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)}} \]

4. Taylor expanded in y around 0 62.0%

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

   1. associate-*r* 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   2. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   3. *-commutative 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x - 1\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   4. sub-neg 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   5. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + \color{blue}{-1}\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + -1\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

6. Simplified 62.0%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

7. Taylor expanded in x around 0 45.6%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\sqrt{5} - 1\right) + \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}} \]
8. Step-by-step derivation

   1. +-commutative 45.6%

      \[\leadsto \frac{0.6666666666666666}{1 + \color{blue}{\left(\left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. fma-def 45.6%

      \[\leadsto \frac{0.6666666666666666}{1 + \color{blue}{\mathsf{fma}\left(1.5 - 0.5 \cdot \sqrt{5}, \cos y, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   3. cancel-sign-sub-inv 45.6%

      \[\leadsto \frac{0.6666666666666666}{1 + \mathsf{fma}\left(\color{blue}{1.5 + \left(-0.5\right) \cdot \sqrt{5}}, \cos y, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)} \]

   4. metadata-eval 45.6%

      \[\leadsto \frac{0.6666666666666666}{1 + \mathsf{fma}\left(1.5 + \color{blue}{-0.5} \cdot \sqrt{5}, \cos y, 0.5 \cdot \left(\sqrt{5} - 1\right)\right)} \]

   5. sub-neg 45.6%

      \[\leadsto \frac{0.6666666666666666}{1 + \mathsf{fma}\left(1.5 + -0.5 \cdot \sqrt{5}, \cos y, 0.5 \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}\right)} \]

   6. metadata-eval 45.6%

      \[\leadsto \frac{0.6666666666666666}{1 + \mathsf{fma}\left(1.5 + -0.5 \cdot \sqrt{5}, \cos y, 0.5 \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right)} \]

9. Applied egg-rr 45.6%

   \[\leadsto \frac{0.6666666666666666}{1 + \color{blue}{\mathsf{fma}\left(1.5 + -0.5 \cdot \sqrt{5}, \cos y, 0.5 \cdot \left(\sqrt{5} + -1\right)\right)}} \]

10. Final simplification 45.6%

    \[\leadsto \frac{0.6666666666666666}{1 + \mathsf{fma}\left(1.5 + \sqrt{5} \cdot -0.5, \cos y, 0.5 \cdot \left(\sqrt{5} + -1\right)\right)} \]

Alternative 22: 42.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{1 + \left(\cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right) + 0.5 \cdot \left(\sqrt{5} + -1\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (+
   1.0
   (+ (* (cos y) (- 1.5 (* 0.5 (sqrt 5.0)))) (* 0.5 (+ (sqrt 5.0) -1.0))))))

C

double code(double x, double y) {
	return 0.6666666666666666 / (1.0 + ((cos(y) * (1.5 - (0.5 * sqrt(5.0)))) + (0.5 * (sqrt(5.0) + -1.0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.6666666666666666d0 / (1.0d0 + ((cos(y) * (1.5d0 - (0.5d0 * sqrt(5.0d0)))) + (0.5d0 * (sqrt(5.0d0) + (-1.0d0)))))
end function

Java

public static double code(double x, double y) {
	return 0.6666666666666666 / (1.0 + ((Math.cos(y) * (1.5 - (0.5 * Math.sqrt(5.0)))) + (0.5 * (Math.sqrt(5.0) + -1.0))));
}

Python

def code(x, y):
	return 0.6666666666666666 / (1.0 + ((math.cos(y) * (1.5 - (0.5 * math.sqrt(5.0)))) + (0.5 * (math.sqrt(5.0) + -1.0))))

Julia

function code(x, y)
	return Float64(0.6666666666666666 / Float64(1.0 + Float64(Float64(cos(y) * Float64(1.5 - Float64(0.5 * sqrt(5.0)))) + Float64(0.5 * Float64(sqrt(5.0) + -1.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.6666666666666666 / (1.0 + ((cos(y) * (1.5 - (0.5 * sqrt(5.0)))) + (0.5 * (sqrt(5.0) + -1.0))));
end

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. cos-neg 99.4%

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

   6. distribute-lft-in 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)}} \]

4. Taylor expanded in y around 0 62.0%

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

   1. associate-*r* 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   2. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   3. *-commutative 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x - 1\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   4. sub-neg 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   5. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + \color{blue}{-1}\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + -1\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

6. Simplified 62.0%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

7. Taylor expanded in x around 0 45.6%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\sqrt{5} - 1\right) + \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}} \]

8. Final simplification 45.6%

   \[\leadsto \frac{0.6666666666666666}{1 + \left(\cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right) + 0.5 \cdot \left(\sqrt{5} + -1\right)\right)} \]

Alternative 23: 40.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{2.5 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) - \sqrt{5}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 0.6666666666666666 (+ 2.5 (* 0.5 (- (+ (sqrt 5.0) -1.0) (sqrt 5.0))))))

C

double code(double x, double y) {
	return 0.6666666666666666 / (2.5 + (0.5 * ((sqrt(5.0) + -1.0) - sqrt(5.0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.6666666666666666d0 / (2.5d0 + (0.5d0 * ((sqrt(5.0d0) + (-1.0d0)) - sqrt(5.0d0))))
end function

Java

public static double code(double x, double y) {
	return 0.6666666666666666 / (2.5 + (0.5 * ((Math.sqrt(5.0) + -1.0) - Math.sqrt(5.0))));
}

Python

def code(x, y):
	return 0.6666666666666666 / (2.5 + (0.5 * ((math.sqrt(5.0) + -1.0) - math.sqrt(5.0))))

Julia

function code(x, y)
	return Float64(0.6666666666666666 / Float64(2.5 + Float64(0.5 * Float64(Float64(sqrt(5.0) + -1.0) - sqrt(5.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.6666666666666666 / (2.5 + (0.5 * ((sqrt(5.0) + -1.0) - sqrt(5.0))));
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(2.5 + N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. cos-neg 99.4%

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

   6. distribute-lft-in 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)}} \]

4. Taylor expanded in y around 0 62.0%

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

   1. associate-*r* 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   2. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   3. *-commutative 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x - 1\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   4. sub-neg 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   5. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + \color{blue}{-1}\right) \cdot \left(\left(-0.0625\right) \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

   6. metadata-eval 62.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\cos x + -1\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin x\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

6. Simplified 62.0%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \color{blue}{\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\cos y, 1.5 - \frac{\sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} + -1}{2}, 1\right)\right)} \]

7. Taylor expanded in x around 0 45.6%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\sqrt{5} - 1\right) + \left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y\right)}} \]

8. Taylor expanded in y around 0 43.9%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\left(2.5 + 0.5 \cdot \left(\sqrt{5} - 1\right)\right) - 0.5 \cdot \sqrt{5}}} \]
9. Step-by-step derivation

   1. associate--l+ 43.9%

      \[\leadsto \frac{0.6666666666666666}{\color{blue}{2.5 + \left(0.5 \cdot \left(\sqrt{5} - 1\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   2. sub-neg 43.9%

      \[\leadsto \frac{0.6666666666666666}{2.5 + \left(0.5 \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} - 0.5 \cdot \sqrt{5}\right)} \]

   3. metadata-eval 43.9%

      \[\leadsto \frac{0.6666666666666666}{2.5 + \left(0.5 \cdot \left(\sqrt{5} + \color{blue}{-1}\right) - 0.5 \cdot \sqrt{5}\right)} \]

   4. distribute-lft-out-- 43.9%

      \[\leadsto \frac{0.6666666666666666}{2.5 + \color{blue}{0.5 \cdot \left(\left(\sqrt{5} + -1\right) - \sqrt{5}\right)}} \]

10. Simplified 43.9%

    \[\leadsto \color{blue}{\frac{0.6666666666666666}{2.5 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) - \sqrt{5}\right)}} \]

11. Final simplification 43.9%

    \[\leadsto \frac{0.6666666666666666}{2.5 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) - \sqrt{5}\right)} \]

Reproduce

herbie shell --seed 2023279 
(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))))))