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

Percentage Accurate: 98.0% → 99.4%

Time: 15.6s

Alternatives: 4

Speedup: 0.7×

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: 98.0% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\left(\frac{x}{y} \cdot \frac{1}{z}\right) \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (*
  0.3333333333333333
  (cbrt
   (pow
    (acos (* (* (/ x y) (/ 1.0 z)) (* 0.05555555555555555 (sqrt t))))
    3.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * cbrt(pow(acos((((x / y) * (1.0 / z)) * (0.05555555555555555 * sqrt(t)))), 3.0));
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * Math.cbrt(Math.pow(Math.acos((((x / y) * (1.0 / z)) * (0.05555555555555555 * Math.sqrt(t)))), 3.0));
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.3333333333333333 * cbrt((acos(Float64(Float64(Float64(x / y) * Float64(1.0 / z)) * Float64(0.05555555555555555 * sqrt(t)))) ^ 3.0)))
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[Power[N[Power[N[ArcCos[N[(N[(N[(x / y), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * N[(0.05555555555555555 * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.1%

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

3. Simplified 98.1%

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

   1. add-cbrt-cube 99.6%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}} \]

   2. pow3 99.6%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{\color{blue}{{\cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}} \]

   3. *-commutative 99.6%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\left(\frac{\frac{x}{y}}{z} \cdot 0.05555555555555555\right)} \cdot \sqrt{t}\right)}^{3}} \]

   4. associate-*l* 99.6%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \color{blue}{\left(\frac{\frac{x}{y}}{z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}}^{3}} \]

   5. associate-/l/ 99.2%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\frac{x}{z \cdot y}} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]

   6. *-commutative 99.2%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\frac{x}{\color{blue}{y \cdot z}} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]

6. Applied egg-rr 99.2%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\sqrt[3]{{\cos^{-1} \left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}}} \]
7. Step-by-step derivation

   1. associate-/r* 99.6%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\frac{\frac{x}{y}}{z}} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]

   2. div-inv 99.6%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{1}{z}\right)} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]

8. Applied egg-rr 99.6%

   \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{1}{z}\right)} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]
9. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\left(0.05555555555555555 \cdot \sqrt{t}\right) \cdot \frac{\frac{x}{z}}{y}\right)}^{3}} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (*
  0.3333333333333333
  (cbrt (pow (acos (* (* 0.05555555555555555 (sqrt t)) (/ (/ x z) y))) 3.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * cbrt(pow(acos(((0.05555555555555555 * sqrt(t)) * ((x / z) / y))), 3.0));
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * Math.cbrt(Math.pow(Math.acos(((0.05555555555555555 * Math.sqrt(t)) * ((x / z) / y))), 3.0));
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.3333333333333333 * cbrt((acos(Float64(Float64(0.05555555555555555 * sqrt(t)) * Float64(Float64(x / z) / y))) ^ 3.0)))
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[Power[N[Power[N[ArcCos[N[(N[(0.05555555555555555 * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.1%

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

3. Simplified 98.1%

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

   1. add-cbrt-cube 99.6%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}} \]

   2. pow3 99.6%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{\color{blue}{{\cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}} \]

   3. *-commutative 99.6%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\left(\frac{\frac{x}{y}}{z} \cdot 0.05555555555555555\right)} \cdot \sqrt{t}\right)}^{3}} \]

   4. associate-*l* 99.6%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \color{blue}{\left(\frac{\frac{x}{y}}{z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}}^{3}} \]

   5. associate-/l/ 99.2%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\frac{x}{z \cdot y}} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]

   6. *-commutative 99.2%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\frac{x}{\color{blue}{y \cdot z}} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]

6. Applied egg-rr 99.2%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\sqrt[3]{{\cos^{-1} \left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}}} \]
7. Step-by-step derivation

   1. associate-/r* 99.6%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\frac{\frac{x}{y}}{z}} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]

   2. div-inv 99.6%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{1}{z}\right)} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]

8. Applied egg-rr 99.6%

   \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\left(\frac{x}{y} \cdot \frac{1}{z}\right)} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]
9. Step-by-step derivation

   1. associate-*l/ 100.0%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\frac{x \cdot \frac{1}{z}}{y}} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]

   2. un-div-inv 100.0%

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\frac{\color{blue}{\frac{x}{z}}}{y} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]

10. Applied egg-rr 100.0%

    \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\color{blue}{\frac{\frac{x}{z}}{y}} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}^{3}} \]

11. Final simplification 100.0%

    \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{{\cos^{-1} \left(\left(0.05555555555555555 \cdot \sqrt{t}\right) \cdot \frac{\frac{x}{z}}{y}\right)}^{3}} \]
12. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.1%

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

3. Simplified 98.1%

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

   1. log1p-expm1-u 98.1%

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

   2. log1p-undefine 98.1%

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

   3. *-commutative 98.1%

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

   4. associate-*l* 98.1%

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

   5. associate-/l/ 97.7%

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

   6. *-commutative 97.7%

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

6. Applied egg-rr 97.7%

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)\right)} \]
7. Step-by-step derivation

   1. log1p-define 97.7%

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

   2. log1p-expm1-u 97.7%

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

   3. rem-cube-cbrt 96.2%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}\right)}^{3}} \]

   4. add-sqr-sqrt 96.8%

      \[\leadsto \color{blue}{\sqrt{0.3333333333333333 \cdot {\left(\sqrt[3]{\cos^{-1} \left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}\right)}^{3}} \cdot \sqrt{0.3333333333333333 \cdot {\left(\sqrt[3]{\cos^{-1} \left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}\right)}^{3}}} \]

   5. sqrt-unprod 96.2%

      \[\leadsto \color{blue}{\sqrt{\left(0.3333333333333333 \cdot {\left(\sqrt[3]{\cos^{-1} \left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}\right)}^{3}\right) \cdot \left(0.3333333333333333 \cdot {\left(\sqrt[3]{\cos^{-1} \left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}\right)}^{3}\right)}} \]

   6. *-commutative 96.2%

      \[\leadsto \sqrt{\color{blue}{\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}\right)}^{3} \cdot 0.3333333333333333\right)} \cdot \left(0.3333333333333333 \cdot {\left(\sqrt[3]{\cos^{-1} \left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}\right)}^{3}\right)} \]

   7. *-commutative 96.2%

      \[\leadsto \sqrt{\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}\right)}^{3} \cdot 0.3333333333333333\right) \cdot \color{blue}{\left({\left(\sqrt[3]{\cos^{-1} \left(\frac{x}{y \cdot z} \cdot \left(0.05555555555555555 \cdot \sqrt{t}\right)\right)}\right)}^{3} \cdot 0.3333333333333333\right)}} \]

8. Applied egg-rr 97.7%

   \[\leadsto \color{blue}{\sqrt{{\cos^{-1} \left(x \cdot \frac{0.05555555555555555 \cdot \sqrt{t}}{y \cdot z}\right)}^{2} \cdot 0.1111111111111111}} \]

10. Applied egg-rr 98.8%

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

    1. sub-neg 98.8%

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

    2. metadata-eval 98.8%

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

    3. +-commutative 98.8%

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

    4. log1p-undefine 96.4%

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

    5. rem-exp-log 96.4%

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

    6. +-commutative 96.4%

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

    7. fma-define 98.8%

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

    8. *-commutative 98.8%

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

    9. associate-*l* 98.8%

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

    10. associate-/l/ 99.2%

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

12. Simplified 99.2%

    \[\leadsto \color{blue}{-1 + \mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(0.05555555555555555 \cdot \left(x \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right), 1\right)} \]
13. Add Preprocessing

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, z, and t 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 y) z))))))

C

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

Fortran

NOTE: x, y, z, and t 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 / y) / z))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, z, and t 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 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.1%

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

3. Simplified 98.1%

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

5. Final simplification 98.1%

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

Developer Target 1: 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 2024172 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (acos (* (/ (/ x 27) (* y z)) (/ (sqrt t) (/ 2 3)))) 3))

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