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

Percentage Accurate: 99.3% → 99.3%

Time: 46.0s

Alternatives: 20

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

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	return (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * Math.pow(Math.cbrt((Math.cos(x) - Math.cos(y))), 3.0))))) / (3.0 + (1.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (Math.cos(y) * (3.0 - Math.sqrt(5.0))))));
}

Julia

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

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in y around inf 99.2%

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

   1. distribute-lft-out 99.2%

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

   2. fma-def 99.2%

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

   3. sub-neg 99.2%

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

   4. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in y around inf 99.2%

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

   1. add-cube-cbrt 99.3%

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

   2. pow3 99.3%

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

9. Applied egg-rr 99.3%

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

10. Final simplification 99.3%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	return (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * Math.cbrt(Math.pow((Math.cos(x) - Math.cos(y)), 3.0)))))) / (3.0 + (1.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (Math.cos(y) * (3.0 - Math.sqrt(5.0))))));
}

Julia

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

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in y around inf 99.2%

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

   1. distribute-lft-out 99.2%

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

   2. fma-def 99.2%

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

   3. sub-neg 99.2%

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

   4. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in y around inf 99.2%

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

   1. add-cbrt-cube 99.2%

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

   2. pow3 99.3%

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

9. Applied egg-rr 99.3%

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

10. Final simplification 99.3%

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

Alternative 3: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in y around inf 99.2%

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

   1. distribute-lft-out 99.2%

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

   2. fma-def 99.2%

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

   3. sub-neg 99.2%

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

   4. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in y around inf 99.2%

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

   1. sub-neg 99.2%

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

   2. metadata-eval 99.2%

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

   3. *-commutative 99.2%

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

   4. fma-def 99.2%

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

9. Applied egg-rr 99.2%

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

10. Final simplification 99.2%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) + ((-0.0625d0) * sin(y))) * ((sin(y) + (sin(x) * (-0.0625d0))) * (cos(x) - cos(y)))))) / (3.0d0 + (1.5d0 * ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) + (cos(y) * (3.0d0 - sqrt(5.0d0))))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.cos(x) - Math.cos(y)))))) / (3.0 + (1.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (Math.cos(y) * (3.0 - Math.sqrt(5.0))))));
}

Python

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.cos(x) - math.cos(y)))))) / (3.0 + (1.5 * ((math.cos(x) * (math.sqrt(5.0) + -1.0)) + (math.cos(y) * (3.0 - math.sqrt(5.0))))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(y)))))) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (cos(y) * (3.0 - sqrt(5.0))))));
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in y around inf 99.2%

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

   1. distribute-lft-out 99.2%

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

   2. fma-def 99.2%

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

   3. sub-neg 99.2%

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

   4. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in y around inf 99.2%

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

8. Final simplification 99.2%

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

Alternative 5: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0125 \lor \neg \left(x \leq 8.5 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{t_1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot t_0 + \cos y \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.0125) (not (<= x 8.5e-20)))
     (/
      (+
       2.0
       (*
        (- (cos x) (cos y))
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) (* (cos y) (/ t_1 2.0)))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (*
         (+ (sin x) (* -0.0625 (sin y)))
         (* (+ (sin y) (* (sin x) -0.0625)) (- 1.0 (cos y))))))
      (+ 3.0 (* 1.5 (+ (* (cos x) t_0) (* (cos y) t_1))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.0125) || !(x <= 8.5e-20)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_1 / 2.0))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((cos(x) * t_0) + (cos(y) * 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(5.0d0) + (-1.0d0)
    t_1 = 3.0d0 - sqrt(5.0d0)
    if ((x <= (-0.0125d0)) .or. (.not. (x <= 8.5d-20))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 / 2.0d0))) + (cos(y) * (t_1 / 2.0d0))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * ((sin(y) + (sin(x) * (-0.0625d0))) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * ((cos(x) * t_0) + (cos(y) * t_1))))
    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 = 3.0 - Math.sqrt(5.0);
	double tmp;
	if ((x <= -0.0125) || !(x <= 8.5e-20)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 / 2.0))) + (Math.cos(y) * (t_1 / 2.0))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (1.0 - Math.cos(y)))))) / (3.0 + (1.5 * ((Math.cos(x) * t_0) + (Math.cos(y) * t_1))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	t_1 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if (x <= -0.0125) or not (x <= 8.5e-20):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 / 2.0))) + (math.cos(y) * (t_1 / 2.0))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * ((math.sin(y) + (math.sin(x) * -0.0625)) * (1.0 - math.cos(y)))))) / (3.0 + (1.5 * ((math.cos(x) * t_0) + (math.cos(y) * t_1))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.0125) || !(x <= 8.5e-20))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(t_1 / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * t_0) + Float64(cos(y) * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	t_1 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if ((x <= -0.0125) || ~((x <= 8.5e-20)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_1 / 2.0))));
	else
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((cos(x) * t_0) + (cos(y) * t_1))));
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0125], N[Not[LessEqual[x, 8.5e-20]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $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[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.012500000000000001 or 8.5000000000000005e-20 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 63.3%

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

   if -0.012500000000000001 < x < 8.5000000000000005e-20

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. fma-def 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around inf 99.6%

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

   8. Taylor expanded in x around 0 99.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0125 \lor \neg \left(x \leq 8.5 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 6: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 1 - \cos y\\ t_2 := 3 - \sqrt{5}\\ t_3 := 3 + 1.5 \cdot \left(\cos x \cdot t_0 + \cos y \cdot t_2\right)\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot t_1\right)\right)}{t_3}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t_0, t_2\right), 1\right)}{\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot t_1\right)\right)}{t_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (- 1.0 (cos y)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (+ 3.0 (* 1.5 (+ (* (cos x) t_0) (* (cos y) t_2))))))
   (if (<= y -4.5e-6)
     (/ (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) t_1)))) t_3)
     (if (<= y 5.8e-12)
       (/
        0.3333333333333333
        (/
         (fma 0.5 (fma (cos x) t_0 t_2) 1.0)
         (fma
          -0.0625
          (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))
          2.0)))
       (/
        (+
         2.0
         (*
          (sqrt 2.0)
          (*
           (+ (sin x) (* -0.0625 (sin y)))
           (* (+ (sin y) (* (sin x) -0.0625)) t_1))))
        t_3)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 1.0 - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 3.0 + (1.5 * ((cos(x) * t_0) + (cos(y) * t_2)));
	double tmp;
	if (y <= -4.5e-6) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * t_1)))) / t_3;
	} else if (y <= 5.8e-12) {
		tmp = 0.3333333333333333 / (fma(0.5, fma(cos(x), t_0, t_2), 1.0) / fma(-0.0625, (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))), 2.0));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * t_1)))) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(1.0 - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * t_0) + Float64(cos(y) * t_2))))
	tmp = 0.0
	if (y <= -4.5e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * t_1)))) / t_3);
	elseif (y <= 5.8e-12)
		tmp = Float64(0.3333333333333333 / Float64(fma(0.5, fma(cos(x), t_0, t_2), 1.0) / fma(-0.0625, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))), 2.0)));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * t_1)))) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.5e-6], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 5.8e-12], N[(0.3333333333333333 / N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] / N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.50000000000000011e-6

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. distribute-lft-out 99.0%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in y around inf 99.1%

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

   8. Taylor expanded in x around 0 67.7%

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

   if -4.50000000000000011e-6 < y < 5.8000000000000003e-12

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0 99.4%

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

      1. associate-*r/ 99.5%

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

      2. associate-/l* 99.6%

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

   4. Simplified 99.6%

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

   if 5.8000000000000003e-12 < y

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. distribute-lft-out 99.0%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in y around inf 99.0%

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

   8. Taylor expanded in x around 0 61.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}{\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 7: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_1 := \cos x - \cos y\\ t_2 := {\sin x}^{2}\\ t_3 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ \mathbf{if}\;x \leq -0.000395:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(t_0 + t_3\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot t_1\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + t_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot t_1\right)\right)}{1 + \left(t_0 \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ (sqrt 5.0) -1.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (pow (sin x) 2.0))
        (t_3 (* (cos y) (- 3.0 (sqrt 5.0)))))
   (if (<= x -0.000395)
     (/
      (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ 3.0 (* 1.5 (+ t_0 t_3))))
     (if (<= x 8.5e-20)
       (/
        (+
         2.0
         (*
          (sqrt 2.0)
          (*
           (+ (sin x) (* -0.0625 (sin y)))
           (* (+ (sin y) (* (sin x) -0.0625)) t_1))))
        (+ 3.0 (* 1.5 (+ -1.0 (+ (sqrt 5.0) t_3)))))
       (*
        0.3333333333333333
        (/
         (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) t_1))))
         (+ 1.0 (+ (* t_0 0.5) (* (cos y) (- 1.5 (* (sqrt 5.0) 0.5)))))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sqrt(5.0) + -1.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = pow(sin(x), 2.0);
	double t_3 = cos(y) * (3.0 - sqrt(5.0));
	double tmp;
	if (x <= -0.000395) {
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * (t_0 + t_3)));
	} else if (x <= 8.5e-20) {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * t_1)))) / (3.0 + (1.5 * (-1.0 + (sqrt(5.0) + t_3))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * t_1)))) / (1.0 + ((t_0 * 0.5) + (cos(y) * (1.5 - (sqrt(5.0) * 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = cos(x) * (sqrt(5.0d0) + (-1.0d0))
    t_1 = cos(x) - cos(y)
    t_2 = sin(x) ** 2.0d0
    t_3 = cos(y) * (3.0d0 - sqrt(5.0d0))
    if (x <= (-0.000395d0)) then
        tmp = (2.0d0 + ((-0.0625d0) * (t_2 * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + (1.5d0 * (t_0 + t_3)))
    else if (x <= 8.5d-20) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * ((sin(y) + (sin(x) * (-0.0625d0))) * t_1)))) / (3.0d0 + (1.5d0 * ((-1.0d0) + (sqrt(5.0d0) + t_3))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (t_2 * (sqrt(2.0d0) * t_1)))) / (1.0d0 + ((t_0 * 0.5d0) + (cos(y) * (1.5d0 - (sqrt(5.0d0) * 0.5d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) * (Math.sqrt(5.0) + -1.0);
	double t_1 = Math.cos(x) - Math.cos(y);
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double t_3 = Math.cos(y) * (3.0 - Math.sqrt(5.0));
	double tmp;
	if (x <= -0.000395) {
		tmp = (2.0 + (-0.0625 * (t_2 * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + (1.5 * (t_0 + t_3)));
	} else if (x <= 8.5e-20) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * t_1)))) / (3.0 + (1.5 * (-1.0 + (Math.sqrt(5.0) + t_3))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_2 * (Math.sqrt(2.0) * t_1)))) / (1.0 + ((t_0 * 0.5) + (Math.cos(y) * (1.5 - (Math.sqrt(5.0) * 0.5))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) * (math.sqrt(5.0) + -1.0)
	t_1 = math.cos(x) - math.cos(y)
	t_2 = math.pow(math.sin(x), 2.0)
	t_3 = math.cos(y) * (3.0 - math.sqrt(5.0))
	tmp = 0
	if x <= -0.000395:
		tmp = (2.0 + (-0.0625 * (t_2 * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + (1.5 * (t_0 + t_3)))
	elif x <= 8.5e-20:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * ((math.sin(y) + (math.sin(x) * -0.0625)) * t_1)))) / (3.0 + (1.5 * (-1.0 + (math.sqrt(5.0) + t_3))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_2 * (math.sqrt(2.0) * t_1)))) / (1.0 + ((t_0 * 0.5) + (math.cos(y) * (1.5 - (math.sqrt(5.0) * 0.5))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(cos(y) * Float64(3.0 - sqrt(5.0)))
	tmp = 0.0
	if (x <= -0.000395)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(1.5 * Float64(t_0 + t_3))));
	elseif (x <= 8.5e-20)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * t_1)))) / Float64(3.0 + Float64(1.5 * Float64(-1.0 + Float64(sqrt(5.0) + t_3)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * t_1)))) / Float64(1.0 + Float64(Float64(t_0 * 0.5) + Float64(cos(y) * Float64(1.5 - Float64(sqrt(5.0) * 0.5)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) * (sqrt(5.0) + -1.0);
	t_1 = cos(x) - cos(y);
	t_2 = sin(x) ^ 2.0;
	t_3 = cos(y) * (3.0 - sqrt(5.0));
	tmp = 0.0;
	if (x <= -0.000395)
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * (t_0 + t_3)));
	elseif (x <= 8.5e-20)
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * t_1)))) / (3.0 + (1.5 * (-1.0 + (sqrt(5.0) + t_3))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * t_1)))) / (1.0 + ((t_0 * 0.5) + (cos(y) * (1.5 - (sqrt(5.0) * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.000395], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-20], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(-1.0 + N[(N[Sqrt[5.0], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(t$95$0 * 0.5), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.95000000000000006e-4

   1. Initial program 98.8%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in y around inf 98.9%

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

   8. Taylor expanded in y around 0 57.7%

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

   if -3.95000000000000006e-4 < x < 8.5000000000000005e-20

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. fma-def 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around inf 99.6%

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

   8. Taylor expanded in x around 0 99.6%

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

   if 8.5000000000000005e-20 < x

   1. Initial program 98.8%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 63.2%

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

   5. Taylor expanded in x around inf 63.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000395:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \end{array} \]

Alternative 8: 79.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{-5} \lor \neg \left(y \leq 5.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot t_1 + \cos y \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t_1, t_0\right), 1\right)}{\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= y -1.05e-5) (not (<= y 5.8e-12)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+ 3.0 (* 1.5 (+ (* (cos x) t_1) (* (cos y) t_0)))))
     (/
      0.3333333333333333
      (/
       (fma 0.5 (fma (cos x) t_1 t_0) 1.0)
       (fma
        -0.0625
        (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))
        2.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -1.05e-5) || !(y <= 5.8e-12)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((cos(x) * t_1) + (cos(y) * t_0))));
	} else {
		tmp = 0.3333333333333333 / (fma(0.5, fma(cos(x), t_1, t_0), 1.0) / fma(-0.0625, (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))), 2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((y <= -1.05e-5) || !(y <= 5.8e-12))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * t_1) + Float64(cos(y) * t_0)))));
	else
		tmp = Float64(0.3333333333333333 / Float64(fma(0.5, fma(cos(x), t_1, t_0), 1.0) / fma(-0.0625, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))), 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[y, -1.05e-5], N[Not[LessEqual[y, 5.8e-12]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.04999999999999994e-5 or 5.8000000000000003e-12 < y

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. distribute-lft-out 99.0%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in y around inf 99.0%

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

   8. Taylor expanded in x around 0 64.1%

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

   if -1.04999999999999994e-5 < y < 5.8000000000000003e-12

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0 99.4%

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

      1. associate-*r/ 99.5%

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

      2. associate-/l* 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-5} \lor \neg \left(y \leq 5.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}{\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}}\\ \end{array} \]

Alternative 9: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{-6} \lor \neg \left(x \leq 8.5 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(t_0 + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos y) (- 3.0 (sqrt 5.0)))))
   (if (or (<= x -8.8e-6) (not (<= x 8.5e-20)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ 3.0 (* 1.5 (+ (* (cos x) (+ (sqrt 5.0) -1.0)) t_0))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+ 3.0 (* 1.5 (+ (sqrt 5.0) (+ t_0 -1.0))))))))

C

double code(double x, double y) {
	double t_0 = cos(y) * (3.0 - sqrt(5.0));
	double tmp;
	if ((x <= -8.8e-6) || !(x <= 8.5e-20)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + (t_0 + -1.0))));
	}
	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 = cos(y) * (3.0d0 - sqrt(5.0d0))
    if ((x <= (-8.8d-6)) .or. (.not. (x <= 8.5d-20))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + (1.5d0 * ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) + t_0)))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * (sqrt(5.0d0) + (t_0 + (-1.0d0)))))
    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));
	double tmp;
	if ((x <= -8.8e-6) || !(x <= 8.5e-20)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + (1.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (1.5 * (Math.sqrt(5.0) + (t_0 + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(y) * (3.0 - math.sqrt(5.0))
	tmp = 0
	if (x <= -8.8e-6) or not (x <= 8.5e-20):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + (1.5 * ((math.cos(x) * (math.sqrt(5.0) + -1.0)) + t_0)))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (1.5 * (math.sqrt(5.0) + (t_0 + -1.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(y) * Float64(3.0 - sqrt(5.0)))
	tmp = 0.0
	if ((x <= -8.8e-6) || !(x <= 8.5e-20))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + t_0))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(sqrt(5.0) + Float64(t_0 + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(y) * (3.0 - sqrt(5.0));
	tmp = 0.0;
	if ((x <= -8.8e-6) || ~((x <= 8.5e-20)))
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + t_0)));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + (t_0 + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -8.8e-6], N[Not[LessEqual[x, 8.5e-20]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.8000000000000004e-6 or 8.5000000000000005e-20 < x

   1. Initial program 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in y around inf 98.8%

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

      1. distribute-lft-out 98.8%

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

      2. fma-def 98.9%

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

      3. sub-neg 98.9%

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

      4. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in y around inf 98.9%

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

   8. Taylor expanded in y around 0 60.1%

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

   if -8.8000000000000004e-6 < x < 8.5000000000000005e-20

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. fma-def 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate--l+ 99.2%

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

   9. Simplified 99.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-6} \lor \neg \left(x \leq 8.5 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(\cos y \cdot \left(3 - \sqrt{5}\right) + -1\right)\right)}\\ \end{array} \]

Alternative 10: 79.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\\ \mathbf{if}\;y \leq -0.00088 \lor \neg \left(y \leq 5.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          3.0
          (*
           1.5
           (+
            (* (cos x) (+ (sqrt 5.0) -1.0))
            (* (cos y) (- 3.0 (sqrt 5.0))))))))
   (if (or (<= y -0.00088) (not (<= y 5.8e-12)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      t_0)
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (cos(y) * (3.0 - sqrt(5.0)))));
	double tmp;
	if ((y <= -0.00088) || !(y <= 5.8e-12)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / 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 = 3.0d0 + (1.5d0 * ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) + (cos(y) * (3.0d0 - sqrt(5.0d0)))))
    if ((y <= (-0.00088d0)) .or. (.not. (y <= 5.8d-12))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / t_0
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (Math.cos(y) * (3.0 - Math.sqrt(5.0)))));
	double tmp;
	if ((y <= -0.00088) || !(y <= 5.8e-12)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / t_0;
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + (1.5 * ((math.cos(x) * (math.sqrt(5.0) + -1.0)) + (math.cos(y) * (3.0 - math.sqrt(5.0)))))
	tmp = 0
	if (y <= -0.00088) or not (y <= 5.8e-12):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / t_0
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))
	tmp = 0.0
	if ((y <= -0.00088) || !(y <= 5.8e-12))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (cos(y) * (3.0 - sqrt(5.0)))));
	tmp = 0.0;
	if ((y <= -0.00088) || ~((y <= 5.8e-12)))
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	else
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.00088], N[Not[LessEqual[y, 5.8e-12]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.80000000000000031e-4 or 5.8000000000000003e-12 < y

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. distribute-lft-out 99.0%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in y around inf 99.0%

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

   8. Taylor expanded in x around 0 63.8%

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

   if -8.80000000000000031e-4 < y < 5.8000000000000003e-12

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in y around inf 99.5%

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

   8. Taylor expanded in y around 0 99.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00088 \lor \neg \left(y \leq 5.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 11: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ t_2 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_3 := \cos x + -1\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_3 \cdot \left(\sqrt{2} \cdot t_1\right)\right)}{3 + 1.5 \cdot \left(3 + \left(t_2 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(\cos y \cdot t_0 + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot t_3\right)\right)}{1 + 0.5 \cdot \left(t_2 + t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (* (cos x) (+ (sqrt 5.0) -1.0)))
        (t_3 (+ (cos x) -1.0)))
   (if (<= x -6.8e-6)
     (/
      (+ 2.0 (* -0.0625 (* t_3 (* (sqrt 2.0) t_1))))
      (+ 3.0 (* 1.5 (+ 3.0 (- t_2 (sqrt 5.0))))))
     (if (<= x 8.5e-20)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (+ (sqrt 5.0) (+ (* (cos y) t_0) -1.0)))))
       (*
        0.3333333333333333
        (/
         (+ 2.0 (* -0.0625 (* t_1 (* (sqrt 2.0) t_3))))
         (+ 1.0 (* 0.5 (+ t_2 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = cos(x) * (sqrt(5.0) + -1.0);
	double t_3 = cos(x) + -1.0;
	double tmp;
	if (x <= -6.8e-6) {
		tmp = (2.0 + (-0.0625 * (t_3 * (sqrt(2.0) * t_1)))) / (3.0 + (1.5 * (3.0 + (t_2 - sqrt(5.0)))));
	} else if (x <= 8.5e-20) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + ((cos(y) * t_0) + -1.0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_1 * (sqrt(2.0) * t_3)))) / (1.0 + (0.5 * (t_2 + 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 3.0d0 - sqrt(5.0d0)
    t_1 = sin(x) ** 2.0d0
    t_2 = cos(x) * (sqrt(5.0d0) + (-1.0d0))
    t_3 = cos(x) + (-1.0d0)
    if (x <= (-6.8d-6)) then
        tmp = (2.0d0 + ((-0.0625d0) * (t_3 * (sqrt(2.0d0) * t_1)))) / (3.0d0 + (1.5d0 * (3.0d0 + (t_2 - sqrt(5.0d0)))))
    else if (x <= 8.5d-20) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * (sqrt(5.0d0) + ((cos(y) * t_0) + (-1.0d0)))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (t_1 * (sqrt(2.0d0) * t_3)))) / (1.0d0 + (0.5d0 * (t_2 + t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = Math.pow(Math.sin(x), 2.0);
	double t_2 = Math.cos(x) * (Math.sqrt(5.0) + -1.0);
	double t_3 = Math.cos(x) + -1.0;
	double tmp;
	if (x <= -6.8e-6) {
		tmp = (2.0 + (-0.0625 * (t_3 * (Math.sqrt(2.0) * t_1)))) / (3.0 + (1.5 * (3.0 + (t_2 - Math.sqrt(5.0)))));
	} else if (x <= 8.5e-20) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (1.5 * (Math.sqrt(5.0) + ((Math.cos(y) * t_0) + -1.0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_1 * (Math.sqrt(2.0) * t_3)))) / (1.0 + (0.5 * (t_2 + t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = math.pow(math.sin(x), 2.0)
	t_2 = math.cos(x) * (math.sqrt(5.0) + -1.0)
	t_3 = math.cos(x) + -1.0
	tmp = 0
	if x <= -6.8e-6:
		tmp = (2.0 + (-0.0625 * (t_3 * (math.sqrt(2.0) * t_1)))) / (3.0 + (1.5 * (3.0 + (t_2 - math.sqrt(5.0)))))
	elif x <= 8.5e-20:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (1.5 * (math.sqrt(5.0) + ((math.cos(y) * t_0) + -1.0))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_1 * (math.sqrt(2.0) * t_3)))) / (1.0 + (0.5 * (t_2 + t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	t_3 = Float64(cos(x) + -1.0)
	tmp = 0.0
	if (x <= -6.8e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_3 * Float64(sqrt(2.0) * t_1)))) / Float64(3.0 + Float64(1.5 * Float64(3.0 + Float64(t_2 - sqrt(5.0))))));
	elseif (x <= 8.5e-20)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(sqrt(5.0) + Float64(Float64(cos(y) * t_0) + -1.0)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_1 * Float64(sqrt(2.0) * t_3)))) / Float64(1.0 + Float64(0.5 * Float64(t_2 + t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = sin(x) ^ 2.0;
	t_2 = cos(x) * (sqrt(5.0) + -1.0);
	t_3 = cos(x) + -1.0;
	tmp = 0.0;
	if (x <= -6.8e-6)
		tmp = (2.0 + (-0.0625 * (t_3 * (sqrt(2.0) * t_1)))) / (3.0 + (1.5 * (3.0 + (t_2 - sqrt(5.0)))));
	elseif (x <= 8.5e-20)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + ((cos(y) * t_0) + -1.0))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_1 * (sqrt(2.0) * t_3)))) / (1.0 + (0.5 * (t_2 + t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -6.8e-6], N[(N[(2.0 + N[(-0.0625 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(3.0 + N[(t$95$2 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-20], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(t$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.80000000000000012e-6

   1. Initial program 98.8%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in y around 0 56.2%

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

      1. sub-neg 56.2%

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

      2. metadata-eval 56.2%

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

      3. associate-*r* 56.2%

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

      4. *-commutative 56.2%

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

      5. associate--l+ 56.2%

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

      6. *-commutative 56.2%

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

      7. sub-neg 56.2%

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

      8. metadata-eval 56.2%

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

   9. Simplified 56.2%

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

   if -6.80000000000000012e-6 < x < 8.5000000000000005e-20

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. fma-def 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate--l+ 99.2%

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

   9. Simplified 99.2%

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

   if 8.5000000000000005e-20 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 61.6%

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

   4. Simplified 61.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(3 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(\cos y \cdot \left(3 - \sqrt{5}\right) + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 12: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\\ t_2 := \cos x + -1\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{3 + 1.5 \cdot \left(3 + t_1\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot t_2\right)\right)}{2.5 + 0.5 \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- (* (cos x) (+ (sqrt 5.0) -1.0)) (sqrt 5.0)))
        (t_2 (+ (cos x) -1.0)))
   (if (<= x -3.5e-5)
     (/
      (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) t_0))))
      (+ 3.0 (* 1.5 (+ 3.0 t_1))))
     (if (<= x 8.5e-20)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (+ -1.0 (+ (sqrt 5.0) (* (cos y) (- 3.0 (sqrt 5.0))))))))
       (*
        0.3333333333333333
        (/
         (+ 2.0 (* -0.0625 (* t_0 (* (sqrt 2.0) t_2))))
         (+ 2.5 (* 0.5 t_1))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = (cos(x) * (sqrt(5.0) + -1.0)) - sqrt(5.0);
	double t_2 = cos(x) + -1.0;
	double tmp;
	if (x <= -3.5e-5) {
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * t_0)))) / (3.0 + (1.5 * (3.0 + t_1)));
	} else if (x <= 8.5e-20) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (-1.0 + (sqrt(5.0) + (cos(y) * (3.0 - sqrt(5.0)))))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * t_2)))) / (2.5 + (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) :: t_2
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    t_1 = (cos(x) * (sqrt(5.0d0) + (-1.0d0))) - sqrt(5.0d0)
    t_2 = cos(x) + (-1.0d0)
    if (x <= (-3.5d-5)) then
        tmp = (2.0d0 + ((-0.0625d0) * (t_2 * (sqrt(2.0d0) * t_0)))) / (3.0d0 + (1.5d0 * (3.0d0 + t_1)))
    else if (x <= 8.5d-20) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * ((-1.0d0) + (sqrt(5.0d0) + (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (t_0 * (sqrt(2.0d0) * t_2)))) / (2.5d0 + (0.5d0 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = (Math.cos(x) * (Math.sqrt(5.0) + -1.0)) - Math.sqrt(5.0);
	double t_2 = Math.cos(x) + -1.0;
	double tmp;
	if (x <= -3.5e-5) {
		tmp = (2.0 + (-0.0625 * (t_2 * (Math.sqrt(2.0) * t_0)))) / (3.0 + (1.5 * (3.0 + t_1)));
	} else if (x <= 8.5e-20) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (1.5 * (-1.0 + (Math.sqrt(5.0) + (Math.cos(y) * (3.0 - Math.sqrt(5.0)))))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * (Math.sqrt(2.0) * t_2)))) / (2.5 + (0.5 * t_1)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = (math.cos(x) * (math.sqrt(5.0) + -1.0)) - math.sqrt(5.0)
	t_2 = math.cos(x) + -1.0
	tmp = 0
	if x <= -3.5e-5:
		tmp = (2.0 + (-0.0625 * (t_2 * (math.sqrt(2.0) * t_0)))) / (3.0 + (1.5 * (3.0 + t_1)))
	elif x <= 8.5e-20:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (1.5 * (-1.0 + (math.sqrt(5.0) + (math.cos(y) * (3.0 - math.sqrt(5.0)))))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * (math.sqrt(2.0) * t_2)))) / (2.5 + (0.5 * t_1)))
	return tmp

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) - sqrt(5.0))
	t_2 = Float64(cos(x) + -1.0)
	tmp = 0.0
	if (x <= -3.5e-5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * t_0)))) / Float64(3.0 + Float64(1.5 * Float64(3.0 + t_1))));
	elseif (x <= 8.5e-20)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(-1.0 + Float64(sqrt(5.0) + Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * t_2)))) / Float64(2.5 + Float64(0.5 * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) ^ 2.0;
	t_1 = (cos(x) * (sqrt(5.0) + -1.0)) - sqrt(5.0);
	t_2 = cos(x) + -1.0;
	tmp = 0.0;
	if (x <= -3.5e-5)
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * t_0)))) / (3.0 + (1.5 * (3.0 + t_1)));
	elseif (x <= 8.5e-20)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (-1.0 + (sqrt(5.0) + (cos(y) * (3.0 - sqrt(5.0)))))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * t_2)))) / (2.5 + (0.5 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -3.5e-5], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(3.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-20], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(-1.0 + N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4999999999999997e-5

   1. Initial program 98.8%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in y around 0 56.2%

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

      1. sub-neg 56.2%

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

      2. metadata-eval 56.2%

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

      3. associate-*r* 56.2%

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

      4. *-commutative 56.2%

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

      5. associate--l+ 56.2%

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

      6. *-commutative 56.2%

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

      7. sub-neg 56.2%

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

      8. metadata-eval 56.2%

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

   9. Simplified 56.2%

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

   if -3.4999999999999997e-5 < x < 8.5000000000000005e-20

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. fma-def 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.2%

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

   if 8.5000000000000005e-20 < x

   1. Initial program 98.8%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 63.2%

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

   5. Taylor expanded in y around 0 61.6%

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

   7. Simplified 61.6%

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

   8. Taylor expanded in x around inf 61.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(3 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\right)}\\ \end{array} \]

Alternative 13: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\\ t_2 := \cos x + -1\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{3 + 1.5 \cdot \left(3 + t_1\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(\cos y \cdot \left(3 - \sqrt{5}\right) + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot t_2\right)\right)}{2.5 + 0.5 \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- (* (cos x) (+ (sqrt 5.0) -1.0)) (sqrt 5.0)))
        (t_2 (+ (cos x) -1.0)))
   (if (<= x -2.65e-5)
     (/
      (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) t_0))))
      (+ 3.0 (* 1.5 (+ 3.0 t_1))))
     (if (<= x 8.5e-20)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (+ (sqrt 5.0) (+ (* (cos y) (- 3.0 (sqrt 5.0))) -1.0)))))
       (*
        0.3333333333333333
        (/
         (+ 2.0 (* -0.0625 (* t_0 (* (sqrt 2.0) t_2))))
         (+ 2.5 (* 0.5 t_1))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = (cos(x) * (sqrt(5.0) + -1.0)) - sqrt(5.0);
	double t_2 = cos(x) + -1.0;
	double tmp;
	if (x <= -2.65e-5) {
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * t_0)))) / (3.0 + (1.5 * (3.0 + t_1)));
	} else if (x <= 8.5e-20) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + ((cos(y) * (3.0 - sqrt(5.0))) + -1.0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * t_2)))) / (2.5 + (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) :: t_2
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    t_1 = (cos(x) * (sqrt(5.0d0) + (-1.0d0))) - sqrt(5.0d0)
    t_2 = cos(x) + (-1.0d0)
    if (x <= (-2.65d-5)) then
        tmp = (2.0d0 + ((-0.0625d0) * (t_2 * (sqrt(2.0d0) * t_0)))) / (3.0d0 + (1.5d0 * (3.0d0 + t_1)))
    else if (x <= 8.5d-20) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * (sqrt(5.0d0) + ((cos(y) * (3.0d0 - sqrt(5.0d0))) + (-1.0d0)))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (t_0 * (sqrt(2.0d0) * t_2)))) / (2.5d0 + (0.5d0 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = (Math.cos(x) * (Math.sqrt(5.0) + -1.0)) - Math.sqrt(5.0);
	double t_2 = Math.cos(x) + -1.0;
	double tmp;
	if (x <= -2.65e-5) {
		tmp = (2.0 + (-0.0625 * (t_2 * (Math.sqrt(2.0) * t_0)))) / (3.0 + (1.5 * (3.0 + t_1)));
	} else if (x <= 8.5e-20) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (1.5 * (Math.sqrt(5.0) + ((Math.cos(y) * (3.0 - Math.sqrt(5.0))) + -1.0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * (Math.sqrt(2.0) * t_2)))) / (2.5 + (0.5 * t_1)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = (math.cos(x) * (math.sqrt(5.0) + -1.0)) - math.sqrt(5.0)
	t_2 = math.cos(x) + -1.0
	tmp = 0
	if x <= -2.65e-5:
		tmp = (2.0 + (-0.0625 * (t_2 * (math.sqrt(2.0) * t_0)))) / (3.0 + (1.5 * (3.0 + t_1)))
	elif x <= 8.5e-20:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (1.5 * (math.sqrt(5.0) + ((math.cos(y) * (3.0 - math.sqrt(5.0))) + -1.0))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * (math.sqrt(2.0) * t_2)))) / (2.5 + (0.5 * t_1)))
	return tmp

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) - sqrt(5.0))
	t_2 = Float64(cos(x) + -1.0)
	tmp = 0.0
	if (x <= -2.65e-5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * t_0)))) / Float64(3.0 + Float64(1.5 * Float64(3.0 + t_1))));
	elseif (x <= 8.5e-20)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(sqrt(5.0) + Float64(Float64(cos(y) * Float64(3.0 - sqrt(5.0))) + -1.0)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * t_2)))) / Float64(2.5 + Float64(0.5 * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) ^ 2.0;
	t_1 = (cos(x) * (sqrt(5.0) + -1.0)) - sqrt(5.0);
	t_2 = cos(x) + -1.0;
	tmp = 0.0;
	if (x <= -2.65e-5)
		tmp = (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * t_0)))) / (3.0 + (1.5 * (3.0 + t_1)));
	elseif (x <= 8.5e-20)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + ((cos(y) * (3.0 - sqrt(5.0))) + -1.0))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * t_2)))) / (2.5 + (0.5 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -2.65e-5], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(3.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-20], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.65e-5

   1. Initial program 98.8%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in y around 0 56.2%

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

      1. sub-neg 56.2%

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

      2. metadata-eval 56.2%

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

      3. associate-*r* 56.2%

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

      4. *-commutative 56.2%

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

      5. associate--l+ 56.2%

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

      6. *-commutative 56.2%

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

      7. sub-neg 56.2%

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

      8. metadata-eval 56.2%

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

   9. Simplified 56.2%

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

   if -2.65e-5 < x < 8.5000000000000005e-20

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. fma-def 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate--l+ 99.2%

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

   9. Simplified 99.2%

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

   if 8.5000000000000005e-20 < x

   1. Initial program 98.8%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 63.2%

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

   5. Taylor expanded in y around 0 61.6%

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

   7. Simplified 61.6%

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

   8. Taylor expanded in x around inf 61.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(3 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(\cos y \cdot \left(3 - \sqrt{5}\right) + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\right)}\\ \end{array} \]

Alternative 14: 60.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (2.5 + (0.5 * ((cos(x) * (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 = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (2.5d0 + (0.5d0 * ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) - sqrt(5.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in y around 0 58.2%

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

5. Taylor expanded in y around 0 55.4%

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

7. Simplified 55.4%

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

8. Taylor expanded in x around inf 55.4%

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

9. Final simplification 55.4%

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

Alternative 15: 60.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (2.0 + (-0.0625 * ((cos(x) + -1.0) * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 + (1.5 * (3.0 + ((cos(x) * (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 = (2.0d0 + ((-0.0625d0) * ((cos(x) + (-1.0d0)) * (sqrt(2.0d0) * (sin(x) ** 2.0d0))))) / (3.0d0 + (1.5d0 * (3.0d0 + ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) - sqrt(5.0d0)))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (-0.0625 * ((Math.cos(x) + -1.0) * (Math.sqrt(2.0) * Math.pow(Math.sin(x), 2.0))))) / (3.0 + (1.5 * (3.0 + ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) - Math.sqrt(5.0)))));
}

Python

def code(x, y):
	return (2.0 + (-0.0625 * ((math.cos(x) + -1.0) * (math.sqrt(2.0) * math.pow(math.sin(x), 2.0))))) / (3.0 + (1.5 * (3.0 + ((math.cos(x) * (math.sqrt(5.0) + -1.0)) - math.sqrt(5.0)))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (-0.0625 * ((cos(x) + -1.0) * (sqrt(2.0) * (sin(x) ^ 2.0))))) / (3.0 + (1.5 * (3.0 + ((cos(x) * (sqrt(5.0) + -1.0)) - sqrt(5.0)))));
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in y around inf 99.2%

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

   1. distribute-lft-out 99.2%

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

   2. fma-def 99.2%

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

   3. sub-neg 99.2%

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

   4. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in y around 0 55.5%

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

   1. sub-neg 55.5%

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

   2. metadata-eval 55.5%

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

   3. associate-*r* 55.5%

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

   4. *-commutative 55.5%

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

   5. associate--l+ 55.5%

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

   6. *-commutative 55.5%

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

   7. sub-neg 55.5%

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

   8. metadata-eval 55.5%

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

9. Simplified 55.5%

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

10. Final simplification 55.5%

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

Alternative 16: 60.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot x\right)}{2}\right)\right)}{\left(2.5 + \cos x \cdot \left(t_0 - 0.5\right)\right) - t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (*
    0.3333333333333333
    (/
     (+
      2.0
      (*
       -0.0625
       (* (* (sqrt 2.0) (+ (cos x) -1.0)) (- 0.5 (/ (cos (* 2.0 x)) 2.0)))))
     (- (+ 2.5 (* (cos x) (- t_0 0.5))) t_0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	return 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * (cos(x) + -1.0)) * (0.5 - (cos((2.0 * x)) / 2.0))))) / ((2.5 + (cos(x) * (t_0 - 0.5))) - 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) * 0.5d0
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sqrt(2.0d0) * (cos(x) + (-1.0d0))) * (0.5d0 - (cos((2.0d0 * x)) / 2.0d0))))) / ((2.5d0 + (cos(x) * (t_0 - 0.5d0))) - t_0))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	return 0.3333333333333333 * ((2.0 + (-0.0625 * ((Math.sqrt(2.0) * (Math.cos(x) + -1.0)) * (0.5 - (Math.cos((2.0 * x)) / 2.0))))) / ((2.5 + (Math.cos(x) * (t_0 - 0.5))) - t_0));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	return 0.3333333333333333 * ((2.0 + (-0.0625 * ((math.sqrt(2.0) * (math.cos(x) + -1.0)) * (0.5 - (math.cos((2.0 * x)) / 2.0))))) / ((2.5 + (math.cos(x) * (t_0 - 0.5))) - t_0))

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * Float64(cos(x) + -1.0)) * Float64(0.5 - Float64(cos(Float64(2.0 * x)) / 2.0))))) / Float64(Float64(2.5 + Float64(cos(x) * Float64(t_0 - 0.5))) - t_0)))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * (cos(x) + -1.0)) * (0.5 - (cos((2.0 * x)) / 2.0))))) / ((2.5 + (cos(x) * (t_0 - 0.5))) - t_0));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.5 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in y around 0 55.4%

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

   1. unpow2 55.4%

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

   2. sin-mult 55.4%

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

6. Applied egg-rr 55.4%

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

   1. div-sub 55.4%

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

   2. +-inverses 55.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

   3. cos-0 55.4%

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

   4. metadata-eval 55.4%

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

   5. count-2 55.4%

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

   6. *-commutative 55.4%

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

8. Simplified 55.4%

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

9. Final simplification 55.4%

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

Alternative 17: 42.4% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (+
   1.0
   (pow
    (cbrt
     (fma (cos y) (+ 1.5 (* (sqrt 5.0) -0.5)) (* (+ (sqrt 5.0) -1.0) 0.5)))
    3.0))))

C

double code(double x, double y) {
	return 0.6666666666666666 / (1.0 + pow(cbrt(fma(cos(y), (1.5 + (sqrt(5.0) * -0.5)), ((sqrt(5.0) + -1.0) * 0.5))), 3.0));
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 / Float64(1.0 + (cbrt(fma(cos(y), Float64(1.5 + Float64(sqrt(5.0) * -0.5)), Float64(Float64(sqrt(5.0) + -1.0) * 0.5))) ^ 3.0)))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(1.0 + N[Power[N[Power[N[(N[Cos[y], $MachinePrecision] * N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in y around 0 58.2%

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

5. Taylor expanded in x around 0 38.7%

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

   1. add-cube-cbrt 38.7%

      \[\leadsto \frac{0.6666666666666666}{1 + \color{blue}{\left(\sqrt[3]{0.5 \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)} \cdot \sqrt[3]{0.5 \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)}\right) \cdot \sqrt[3]{0.5 \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)}}} \]

   2. pow3 38.7%

      \[\leadsto \frac{0.6666666666666666}{1 + \color{blue}{{\left(\sqrt[3]{0.5 \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)}\right)}^{3}}} \]

7. Applied egg-rr 38.7%

   \[\leadsto \frac{0.6666666666666666}{1 + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos y, 1.5 + -0.5 \cdot \sqrt{5}, 0.5 \cdot \left(\sqrt{5} + -1\right)\right)}\right)}^{3}}} \]

8. Final simplification 38.7%

   \[\leadsto \frac{0.6666666666666666}{1 + {\left(\sqrt[3]{\mathsf{fma}\left(\cos y, 1.5 + \sqrt{5} \cdot -0.5, \left(\sqrt{5} + -1\right) \cdot 0.5\right)}\right)}^{3}} \]

Alternative 18: 42.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in y around 0 58.2%

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

5. Taylor expanded in x around 0 38.7%

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

   1. div-inv 38.7%

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

   2. +-commutative 38.7%

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

   3. *-commutative 38.7%

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

   4. fma-def 38.7%

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

   5. *-commutative 38.7%

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

   6. cancel-sign-sub-inv 38.7%

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

   7. metadata-eval 38.7%

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

   8. sub-neg 38.7%

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

   9. metadata-eval 38.7%

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

7. Applied egg-rr 38.7%

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

8. Final simplification 38.7%

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

Alternative 19: 42.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

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 - (sqrt(5.0d0) * 0.5d0))) + ((sqrt(5.0d0) + (-1.0d0)) * 0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in y around 0 58.2%

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

5. Taylor expanded in x around 0 38.7%

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

6. Final simplification 38.7%

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

Alternative 20: 40.4% accurate, 1139.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in y around 0 58.2%

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

5. Taylor expanded in y around 0 55.4%

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

7. Simplified 55.4%

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

8. Taylor expanded in x around 0 36.6%

   \[\leadsto \color{blue}{0.3333333333333333} \]

9. Final simplification 36.6%

   \[\leadsto 0.3333333333333333 \]

Reproduce

herbie shell --seed 2023318 
(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))))))