Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, D

Percentage Accurate: 98.1% → 99.5%

Time: 10.6s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))

C

double code(double x, double y, double z, double t) {
	return (1.0 / 3.0) * acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (1.0d0 / 3.0d0) * acos((((3.0d0 * (x / (y * 27.0d0))) / (z * 2.0d0)) * sqrt(t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (1.0 / 3.0) * Math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * Math.sqrt(t)));
}

Python

def code(x, y, z, t):
	return (1.0 / 3.0) * math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * math.sqrt(t)))

Julia

function code(x, y, z, t)
	return Float64(Float64(1.0 / 3.0) * acos(Float64(Float64(Float64(3.0 * Float64(x / Float64(y * 27.0))) / Float64(z * 2.0)) * sqrt(t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (1.0 / 3.0) * acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(N[(N[(3.0 * N[(x / N[(y * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))

C

double code(double x, double y, double z, double t) {
	return (1.0 / 3.0) * acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (1.0d0 / 3.0d0) * acos((((3.0d0 * (x / (y * 27.0d0))) / (z * 2.0d0)) * sqrt(t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (1.0 / 3.0) * Math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * Math.sqrt(t)));
}

Python

def code(x, y, z, t):
	return (1.0 / 3.0) * math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * math.sqrt(t)))

Julia

function code(x, y, z, t)
	return Float64(Float64(1.0 / 3.0) * acos(Float64(Float64(Float64(3.0 * Float64(x / Float64(y * 27.0))) / Float64(z * 2.0)) * sqrt(t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (1.0 / 3.0) * acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(N[(N[(3.0 * N[(x / N[(y * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [y, z] = \mathsf{sort}([y, z])\\ \\ e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{\frac{0.05555555555555555 \cdot x}{y}}{z}\right)\right)} + -1 \end{array} \]

FPCore

NOTE: y and z should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+
  (exp
   (log1p
    (*
     0.3333333333333333
     (acos (* (sqrt t) (/ (/ (* 0.05555555555555555 x) y) z))))))
  -1.0))

C

assert(y < z);
double code(double x, double y, double z, double t) {
	return exp(log1p((0.3333333333333333 * acos((sqrt(t) * (((0.05555555555555555 * x) / y) / z)))))) + -1.0;
}

Java

assert y < z;
public static double code(double x, double y, double z, double t) {
	return Math.exp(Math.log1p((0.3333333333333333 * Math.acos((Math.sqrt(t) * (((0.05555555555555555 * x) / y) / z)))))) + -1.0;
}

Python

[y, z] = sort([y, z])
def code(x, y, z, t):
	return math.exp(math.log1p((0.3333333333333333 * math.acos((math.sqrt(t) * (((0.05555555555555555 * x) / y) / z)))))) + -1.0

Julia

y, z = sort([y, z])
function code(x, y, z, t)
	return Float64(exp(log1p(Float64(0.3333333333333333 * acos(Float64(sqrt(t) * Float64(Float64(Float64(0.05555555555555555 * x) / y) / z)))))) + -1.0)
end

Wolfram

NOTE: y and z should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Exp[N[Log[1 + N[(0.3333333333333333 * N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] * N[(N[(N[(0.05555555555555555 * x), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 97.7%

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

3. Simplified 97.7%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{3}{z} \cdot \frac{\frac{x}{y}}{54}\right) \cdot \sqrt{t}\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 97.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{3}{z} \cdot \frac{\frac{x}{y}}{54}\right) \cdot \sqrt{t}\right)\right)\right)} \]

   2. expm1-udef 99.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{3}{z} \cdot \frac{\frac{x}{y}}{54}\right) \cdot \sqrt{t}\right)\right)} - 1} \]

5. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{x}{\left(z \cdot 0.3333333333333333\right) \cdot \left(y \cdot 54\right)}\right)\right)} - 1} \]

6. Taylor expanded in x around 0 99.0%

   \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \color{blue}{\left(0.05555555555555555 \cdot \frac{x}{y \cdot z}\right)}\right)\right)} - 1 \]
7. Step-by-step derivation

   1. associate-*r/ 99.0%

      \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \color{blue}{\frac{0.05555555555555555 \cdot x}{y \cdot z}}\right)\right)} - 1 \]

   2. associate-/r* 99.2%

      \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \color{blue}{\frac{\frac{0.05555555555555555 \cdot x}{y}}{z}}\right)\right)} - 1 \]

8. Simplified 99.2%

   \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \color{blue}{\frac{\frac{0.05555555555555555 \cdot x}{y}}{z}}\right)\right)} - 1 \]

9. Final simplification 99.2%

   \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{\frac{0.05555555555555555 \cdot x}{y}}{z}\right)\right)} + -1 \]

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [y, z] = \mathsf{sort}([y, z])\\ \\ e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \frac{\sqrt{t} \cdot x}{y \cdot z}\right)\right)} + -1 \end{array} \]

FPCore

NOTE: y and z should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+
  (exp
   (log1p
    (*
     0.3333333333333333
     (acos (* 0.05555555555555555 (/ (* (sqrt t) x) (* y z)))))))
  -1.0))

C

assert(y < z);
double code(double x, double y, double z, double t) {
	return exp(log1p((0.3333333333333333 * acos((0.05555555555555555 * ((sqrt(t) * x) / (y * z))))))) + -1.0;
}

Java

assert y < z;
public static double code(double x, double y, double z, double t) {
	return Math.exp(Math.log1p((0.3333333333333333 * Math.acos((0.05555555555555555 * ((Math.sqrt(t) * x) / (y * z))))))) + -1.0;
}

Python

[y, z] = sort([y, z])
def code(x, y, z, t):
	return math.exp(math.log1p((0.3333333333333333 * math.acos((0.05555555555555555 * ((math.sqrt(t) * x) / (y * z))))))) + -1.0

Julia

y, z = sort([y, z])
function code(x, y, z, t)
	return Float64(exp(log1p(Float64(0.3333333333333333 * acos(Float64(0.05555555555555555 * Float64(Float64(sqrt(t) * x) / Float64(y * z))))))) + -1.0)
end

Wolfram

NOTE: y and z should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Exp[N[Log[1 + N[(0.3333333333333333 * N[ArcCos[N[(0.05555555555555555 * N[(N[(N[Sqrt[t], $MachinePrecision] * x), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 97.7%

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
2. Step-by-step derivation

   1. metadata-eval 97.7%

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   2. *-commutative 97.7%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{\color{blue}{2 \cdot z}} \cdot \sqrt{t}\right) \]

   3. times-frac 97.7%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\color{blue}{\left(\frac{3}{2} \cdot \frac{\frac{x}{y \cdot 27}}{z}\right)} \cdot \sqrt{t}\right) \]

   4. associate-*l* 97.7%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{3}{2} \cdot \left(\frac{\frac{x}{y \cdot 27}}{z} \cdot \sqrt{t}\right)\right)} \]

   5. associate-/l/ 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\color{blue}{\frac{x}{z \cdot \left(y \cdot 27\right)}} \cdot \sqrt{t}\right)\right) \]

   6. *-commutative 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\frac{x}{z \cdot \color{blue}{\left(27 \cdot y\right)}} \cdot \sqrt{t}\right)\right) \]

   7. associate-*r* 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\frac{x}{\color{blue}{\left(z \cdot 27\right) \cdot y}} \cdot \sqrt{t}\right)\right) \]

   8. *-commutative 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\frac{x}{\color{blue}{y \cdot \left(z \cdot 27\right)}} \cdot \sqrt{t}\right)\right) \]

   9. associate-/l/ 98.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\color{blue}{\frac{\frac{x}{z \cdot 27}}{y}} \cdot \sqrt{t}\right)\right) \]

   10. associate-*l* 98.5%

       \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\left(\frac{3}{2} \cdot \frac{\frac{x}{z \cdot 27}}{y}\right) \cdot \sqrt{t}\right)} \]

   11. times-frac 98.5%

       \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\color{blue}{\frac{3 \cdot \frac{x}{z \cdot 27}}{2 \cdot y}} \cdot \sqrt{t}\right) \]

   12. *-commutative 98.5%

       \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{z \cdot 27}}{\color{blue}{y \cdot 2}} \cdot \sqrt{t}\right) \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{z}}{y}\right) \cdot \sqrt{t}\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 98.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{z}}{y}\right) \cdot \sqrt{t}\right)\right)\right)} \]

   2. expm1-udef 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{z}}{y}\right) \cdot \sqrt{t}\right)\right)} - 1} \]

   3. associate-*l* 100.0%

      \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(0.05555555555555555 \cdot \left(\frac{\frac{x}{z}}{y} \cdot \sqrt{t}\right)\right)}\right)} - 1 \]

   4. associate-/l/ 99.0%

      \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \left(\color{blue}{\frac{x}{y \cdot z}} \cdot \sqrt{t}\right)\right)\right)} - 1 \]

   5. associate-*l/ 98.4%

      \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \color{blue}{\frac{x \cdot \sqrt{t}}{y \cdot z}}\right)\right)} - 1 \]

   6. *-commutative 98.4%

      \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \frac{x \cdot \sqrt{t}}{\color{blue}{z \cdot y}}\right)\right)} - 1 \]

5. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \frac{x \cdot \sqrt{t}}{z \cdot y}\right)\right)} - 1} \]

6. Final simplification 98.4%

   \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \frac{\sqrt{t} \cdot x}{y \cdot z}\right)\right)} + -1 \]

Alternative 3: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, z] = \mathsf{sort}([y, z])\\ \\ 0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{\frac{z}{\frac{x}{y}}}\right) \end{array} \]

FPCore

NOTE: y and z should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (*
  0.3333333333333333
  (acos (* 0.05555555555555555 (/ (sqrt t) (/ z (/ x y)))))))

C

assert(y < z);
double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * acos((0.05555555555555555 * (sqrt(t) / (z / (x / y)))));
}

Fortran

NOTE: y and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.3333333333333333d0 * acos((0.05555555555555555d0 * (sqrt(t) / (z / (x / y)))))
end function

Java

assert y < z;
public static double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * Math.acos((0.05555555555555555 * (Math.sqrt(t) / (z / (x / y)))));
}

Python

[y, z] = sort([y, z])
def code(x, y, z, t):
	return 0.3333333333333333 * math.acos((0.05555555555555555 * (math.sqrt(t) / (z / (x / y)))))

Julia

y, z = sort([y, z])
function code(x, y, z, t)
	return Float64(0.3333333333333333 * acos(Float64(0.05555555555555555 * Float64(sqrt(t) / Float64(z / Float64(x / y))))))
end

MATLAB

y, z = num2cell(sort([y, z])){:}
function tmp = code(x, y, z, t)
	tmp = 0.3333333333333333 * acos((0.05555555555555555 * (sqrt(t) / (z / (x / y)))));
end

Wolfram

NOTE: y and z should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[ArcCos[N[(0.05555555555555555 * N[(N[Sqrt[t], $MachinePrecision] / N[(z / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.7%

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
2. Step-by-step derivation

   1. metadata-eval 97.7%

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   2. *-commutative 97.7%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{\color{blue}{2 \cdot z}} \cdot \sqrt{t}\right) \]

   3. times-frac 97.7%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\color{blue}{\left(\frac{3}{2} \cdot \frac{\frac{x}{y \cdot 27}}{z}\right)} \cdot \sqrt{t}\right) \]

   4. associate-*l* 97.7%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{3}{2} \cdot \left(\frac{\frac{x}{y \cdot 27}}{z} \cdot \sqrt{t}\right)\right)} \]

   5. associate-/l/ 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\color{blue}{\frac{x}{z \cdot \left(y \cdot 27\right)}} \cdot \sqrt{t}\right)\right) \]

   6. *-commutative 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\frac{x}{z \cdot \color{blue}{\left(27 \cdot y\right)}} \cdot \sqrt{t}\right)\right) \]

   7. associate-*r* 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\frac{x}{\color{blue}{\left(z \cdot 27\right) \cdot y}} \cdot \sqrt{t}\right)\right) \]

   8. *-commutative 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\frac{x}{\color{blue}{y \cdot \left(z \cdot 27\right)}} \cdot \sqrt{t}\right)\right) \]

   9. associate-/l/ 98.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\color{blue}{\frac{\frac{x}{z \cdot 27}}{y}} \cdot \sqrt{t}\right)\right) \]

   10. associate-*l* 98.5%

       \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\left(\frac{3}{2} \cdot \frac{\frac{x}{z \cdot 27}}{y}\right) \cdot \sqrt{t}\right)} \]

   11. times-frac 98.5%

       \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\color{blue}{\frac{3 \cdot \frac{x}{z \cdot 27}}{2 \cdot y}} \cdot \sqrt{t}\right) \]

   12. *-commutative 98.5%

       \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{z \cdot 27}}{\color{blue}{y \cdot 2}} \cdot \sqrt{t}\right) \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{z}}{y}\right) \cdot \sqrt{t}\right)} \]

4. Taylor expanded in x around 0 97.5%

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(0.05555555555555555 \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r/ 96.9%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \color{blue}{\frac{\sqrt{t} \cdot x}{y \cdot z}}\right) \]

   2. *-commutative 96.9%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \frac{\sqrt{t} \cdot x}{\color{blue}{z \cdot y}}\right) \]

   3. associate-/l* 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \color{blue}{\frac{\sqrt{t}}{\frac{z \cdot y}{x}}}\right) \]

   4. associate-/l* 97.7%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{\color{blue}{\frac{z}{\frac{x}{y}}}}\right) \]

6. Simplified 97.7%

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(0.05555555555555555 \cdot \frac{\sqrt{t}}{\frac{z}{\frac{x}{y}}}\right)} \]

7. Final simplification 97.7%

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{\frac{z}{\frac{x}{y}}}\right) \]

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [y, z] = \mathsf{sort}([y, z])\\ \\ 0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{z}}{y}\right)\right) \end{array} \]

FPCore

NOTE: y and z should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (*
  0.3333333333333333
  (acos (* (sqrt t) (* 0.05555555555555555 (/ (/ x z) y))))))

C

assert(y < z);
double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * acos((sqrt(t) * (0.05555555555555555 * ((x / z) / y))));
}

Fortran

NOTE: y and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.3333333333333333d0 * acos((sqrt(t) * (0.05555555555555555d0 * ((x / z) / y))))
end function

Java

assert y < z;
public static double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * Math.acos((Math.sqrt(t) * (0.05555555555555555 * ((x / z) / y))));
}

Python

[y, z] = sort([y, z])
def code(x, y, z, t):
	return 0.3333333333333333 * math.acos((math.sqrt(t) * (0.05555555555555555 * ((x / z) / y))))

Julia

y, z = sort([y, z])
function code(x, y, z, t)
	return Float64(0.3333333333333333 * acos(Float64(sqrt(t) * Float64(0.05555555555555555 * Float64(Float64(x / z) / y)))))
end

MATLAB

y, z = num2cell(sort([y, z])){:}
function tmp = code(x, y, z, t)
	tmp = 0.3333333333333333 * acos((sqrt(t) * (0.05555555555555555 * ((x / z) / y))));
end

Wolfram

NOTE: y and z should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] * N[(0.05555555555555555 * N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.7%

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
2. Step-by-step derivation

   1. metadata-eval 97.7%

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   2. *-commutative 97.7%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{\color{blue}{2 \cdot z}} \cdot \sqrt{t}\right) \]

   3. times-frac 97.7%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\color{blue}{\left(\frac{3}{2} \cdot \frac{\frac{x}{y \cdot 27}}{z}\right)} \cdot \sqrt{t}\right) \]

   4. associate-*l* 97.7%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{3}{2} \cdot \left(\frac{\frac{x}{y \cdot 27}}{z} \cdot \sqrt{t}\right)\right)} \]

   5. associate-/l/ 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\color{blue}{\frac{x}{z \cdot \left(y \cdot 27\right)}} \cdot \sqrt{t}\right)\right) \]

   6. *-commutative 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\frac{x}{z \cdot \color{blue}{\left(27 \cdot y\right)}} \cdot \sqrt{t}\right)\right) \]

   7. associate-*r* 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\frac{x}{\color{blue}{\left(z \cdot 27\right) \cdot y}} \cdot \sqrt{t}\right)\right) \]

   8. *-commutative 97.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\frac{x}{\color{blue}{y \cdot \left(z \cdot 27\right)}} \cdot \sqrt{t}\right)\right) \]

   9. associate-/l/ 98.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{2} \cdot \left(\color{blue}{\frac{\frac{x}{z \cdot 27}}{y}} \cdot \sqrt{t}\right)\right) \]

   10. associate-*l* 98.5%

       \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\left(\frac{3}{2} \cdot \frac{\frac{x}{z \cdot 27}}{y}\right) \cdot \sqrt{t}\right)} \]

   11. times-frac 98.5%

       \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\color{blue}{\frac{3 \cdot \frac{x}{z \cdot 27}}{2 \cdot y}} \cdot \sqrt{t}\right) \]

   12. *-commutative 98.5%

       \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{z \cdot 27}}{\color{blue}{y \cdot 2}} \cdot \sqrt{t}\right) \]

3. Simplified 98.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{z}}{y}\right) \cdot \sqrt{t}\right)} \]

4. Final simplification 98.5%

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{z}}{y}\right)\right) \]

Developer target: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (acos (* (/ (/ x 27.0) (* y z)) (/ (sqrt t) (/ 2.0 3.0)))) 3.0))

C

double code(double x, double y, double z, double t) {
	return acos((((x / 27.0) / (y * z)) * (sqrt(t) / (2.0 / 3.0)))) / 3.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = acos((((x / 27.0d0) / (y * z)) * (sqrt(t) / (2.0d0 / 3.0d0)))) / 3.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.acos((((x / 27.0) / (y * z)) * (Math.sqrt(t) / (2.0 / 3.0)))) / 3.0;
}

Python

def code(x, y, z, t):
	return math.acos((((x / 27.0) / (y * z)) * (math.sqrt(t) / (2.0 / 3.0)))) / 3.0

Julia

function code(x, y, z, t)
	return Float64(acos(Float64(Float64(Float64(x / 27.0) / Float64(y * z)) * Float64(sqrt(t) / Float64(2.0 / 3.0)))) / 3.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = acos((((x / 27.0) / (y * z)) * (sqrt(t) / (2.0 / 3.0)))) / 3.0;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[ArcCos[N[(N[(N[(x / 27.0), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t], $MachinePrecision] / N[(2.0 / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27.0) (* y z)) (/ (sqrt t) (/ 2.0 3.0)))) 3.0)

  (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))