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

Percentage Accurate: 97.9% → 99.4%

Time: 14.7s

Alternatives: 3

Speedup: 0.5×

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: 97.9% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(exp(log1p(Float64(0.3333333333333333 * acos(Float64(Float64(Float64(x / y) / z) * Float64(0.05555555555555555 * sqrt(t))))))) + -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[(N[Exp[N[Log[1 + N[(0.3333333333333333 * N[ArcCos[N[(N[(N[(x / y), $MachinePrecision] / z), $MachinePrecision] * N[(0.05555555555555555 * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 97.7%

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

3. Simplified 97.7%

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

   1. expm1-log1p-u 97.7%

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

   2. expm1-undefine 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*l* 99.2%

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

   5. associate-/l/ 99.2%

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

   6. *-commutative 99.2%

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

6. Applied egg-rr 99.2%

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

   1. div-inv 98.8%

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

8. Applied egg-rr 98.8%

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

   1. associate-*r/ 99.2%

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

   2. *-rgt-identity 99.2%

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

   3. associate-/r* 99.2%

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

10. Simplified 99.2%

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

11. Final simplification 99.2%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

assert(x < y && y < z && z < t);
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 x < y && y < z && z < t;
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

[x, y, z, t] = sort([x, y, z, t])
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

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(exp(log1p(Float64(0.3333333333333333 * acos(Float64(Float64(0.05555555555555555 * sqrt(t)) * Float64(x / 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[(N[Exp[N[Log[1 + N[(0.3333333333333333 * N[ArcCos[N[(N[(0.05555555555555555 * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 97.7%

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

3. Simplified 97.7%

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

   1. expm1-log1p-u 97.7%

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

   2. expm1-undefine 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*l* 99.2%

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

   5. associate-/l/ 99.2%

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

   6. *-commutative 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

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) (* (/ (/ x y) z) 0.05555555555555555)))))

C

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

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) * (((x / y) / z) * 0.05555555555555555d0)))
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) * (((x / y) / z) * 0.05555555555555555)));
}

Python

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

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(Float64(Float64(x / y) / z) * 0.05555555555555555))))
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) * (((x / y) / z) * 0.05555555555555555)));
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[(N[(N[(x / y), $MachinePrecision] / z), $MachinePrecision] * 0.05555555555555555), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.7%

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

3. Simplified 97.7%

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

5. Final simplification 97.7%

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

Developer target: 98.2% 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 2024082 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :alt
  (/ (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)))))