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

Percentage Accurate: 98.0% → 99.5%

Time: 26.9s

Alternatives: 5

Speedup: 0.2×

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: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\pi}{\frac{4}{\pi}}\\ t_2 := \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right)\\ \frac{0.3333333333333333}{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} + {t\_2}^{3}} \cdot \frac{t\_1 - {t\_2}^{2}}{\frac{1}{t\_1 + t\_2 \cdot \left(t\_2 + \frac{\pi}{-2}\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ PI (/ 4.0 PI)))
        (t_2 (asin (/ (sqrt t) (/ z (/ x (* 18.0 y)))))))
   (*
    (/ 0.3333333333333333 (+ (/ (* PI (* PI PI)) 8.0) (pow t_2 3.0)))
    (/ (- t_1 (pow t_2 2.0)) (/ 1.0 (+ t_1 (* t_2 (+ t_2 (/ PI -2.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((double) M_PI) / (4.0 / ((double) M_PI));
	double t_2 = asin((sqrt(t) / (z / (x / (18.0 * y)))));
	return (0.3333333333333333 / (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) / 8.0) + pow(t_2, 3.0))) * ((t_1 - pow(t_2, 2.0)) / (1.0 / (t_1 + (t_2 * (t_2 + (((double) M_PI) / -2.0))))));
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.PI / (4.0 / Math.PI);
	double t_2 = Math.asin((Math.sqrt(t) / (z / (x / (18.0 * y)))));
	return (0.3333333333333333 / (((Math.PI * (Math.PI * Math.PI)) / 8.0) + Math.pow(t_2, 3.0))) * ((t_1 - Math.pow(t_2, 2.0)) / (1.0 / (t_1 + (t_2 * (t_2 + (Math.PI / -2.0))))));
}

Python

def code(x, y, z, t):
	t_1 = math.pi / (4.0 / math.pi)
	t_2 = math.asin((math.sqrt(t) / (z / (x / (18.0 * y)))))
	return (0.3333333333333333 / (((math.pi * (math.pi * math.pi)) / 8.0) + math.pow(t_2, 3.0))) * ((t_1 - math.pow(t_2, 2.0)) / (1.0 / (t_1 + (t_2 * (t_2 + (math.pi / -2.0))))))

Julia

function code(x, y, z, t)
	t_1 = Float64(pi / Float64(4.0 / pi))
	t_2 = asin(Float64(sqrt(t) / Float64(z / Float64(x / Float64(18.0 * y)))))
	return Float64(Float64(0.3333333333333333 / Float64(Float64(Float64(pi * Float64(pi * pi)) / 8.0) + (t_2 ^ 3.0))) * Float64(Float64(t_1 - (t_2 ^ 2.0)) / Float64(1.0 / Float64(t_1 + Float64(t_2 * Float64(t_2 + Float64(pi / -2.0)))))))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = pi / (4.0 / pi);
	t_2 = asin((sqrt(t) / (z / (x / (18.0 * y)))));
	tmp = (0.3333333333333333 / (((pi * (pi * pi)) / 8.0) + (t_2 ^ 3.0))) * ((t_1 - (t_2 ^ 2.0)) / (1.0 / (t_1 + (t_2 * (t_2 + (pi / -2.0))))));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(Pi / N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcSin[N[(N[Sqrt[t], $MachinePrecision] / N[(z / N[(x / N[(18.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(0.3333333333333333 / N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / 8.0), $MachinePrecision] + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 - N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(t$95$1 + N[(t$95$2 * N[(t$95$2 + N[(Pi / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.5%

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right), \color{blue}{\frac{1}{3}}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\sqrt{t}}{z \cdot \frac{18}{\frac{x}{y}}}\right) \cdot 0.3333333333333333} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} + {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right)}^{3}} \cdot \frac{\frac{\pi}{\frac{4}{\pi}} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right)}^{2}}{\frac{1}{\frac{\pi}{\frac{4}{\pi}} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right) \cdot \left(\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right) + \frac{\pi}{-2}\right)}}} \]
7. Add Preprocessing

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\pi}{\frac{4}{\pi}}\\ t_2 := \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{x}{18}}{y}}}\right)\\ \frac{0.3333333333333333}{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} + {t\_2}^{3}} \cdot \left(\left(t\_1 - {t\_2}^{2}\right) \cdot \left(t\_1 + t\_2 \cdot \left(\frac{\pi}{-2} + t\_2\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ PI (/ 4.0 PI)))
        (t_2 (asin (/ (sqrt t) (/ z (/ (/ x 18.0) y))))))
   (*
    (/ 0.3333333333333333 (+ (/ (* PI (* PI PI)) 8.0) (pow t_2 3.0)))
    (* (- t_1 (pow t_2 2.0)) (+ t_1 (* t_2 (+ (/ PI -2.0) t_2)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((double) M_PI) / (4.0 / ((double) M_PI));
	double t_2 = asin((sqrt(t) / (z / ((x / 18.0) / y))));
	return (0.3333333333333333 / (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) / 8.0) + pow(t_2, 3.0))) * ((t_1 - pow(t_2, 2.0)) * (t_1 + (t_2 * ((((double) M_PI) / -2.0) + t_2))));
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.PI / (4.0 / Math.PI);
	double t_2 = Math.asin((Math.sqrt(t) / (z / ((x / 18.0) / y))));
	return (0.3333333333333333 / (((Math.PI * (Math.PI * Math.PI)) / 8.0) + Math.pow(t_2, 3.0))) * ((t_1 - Math.pow(t_2, 2.0)) * (t_1 + (t_2 * ((Math.PI / -2.0) + t_2))));
}

Python

def code(x, y, z, t):
	t_1 = math.pi / (4.0 / math.pi)
	t_2 = math.asin((math.sqrt(t) / (z / ((x / 18.0) / y))))
	return (0.3333333333333333 / (((math.pi * (math.pi * math.pi)) / 8.0) + math.pow(t_2, 3.0))) * ((t_1 - math.pow(t_2, 2.0)) * (t_1 + (t_2 * ((math.pi / -2.0) + t_2))))

Julia

function code(x, y, z, t)
	t_1 = Float64(pi / Float64(4.0 / pi))
	t_2 = asin(Float64(sqrt(t) / Float64(z / Float64(Float64(x / 18.0) / y))))
	return Float64(Float64(0.3333333333333333 / Float64(Float64(Float64(pi * Float64(pi * pi)) / 8.0) + (t_2 ^ 3.0))) * Float64(Float64(t_1 - (t_2 ^ 2.0)) * Float64(t_1 + Float64(t_2 * Float64(Float64(pi / -2.0) + t_2)))))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = pi / (4.0 / pi);
	t_2 = asin((sqrt(t) / (z / ((x / 18.0) / y))));
	tmp = (0.3333333333333333 / (((pi * (pi * pi)) / 8.0) + (t_2 ^ 3.0))) * ((t_1 - (t_2 ^ 2.0)) * (t_1 + (t_2 * ((pi / -2.0) + t_2))));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(Pi / N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcSin[N[(N[Sqrt[t], $MachinePrecision] / N[(z / N[(N[(x / 18.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(0.3333333333333333 / N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / 8.0), $MachinePrecision] + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 - N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[(t$95$2 * N[(N[(Pi / -2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 98.5%

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right), \color{blue}{\frac{1}{3}}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\sqrt{t}}{z \cdot \frac{18}{\frac{x}{y}}}\right) \cdot 0.3333333333333333} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} + {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right)}^{3}} \cdot \frac{\frac{\pi}{\frac{4}{\pi}} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right)}^{2}}{\frac{1}{\frac{\pi}{\frac{4}{\pi}} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right) \cdot \left(\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right) + \frac{\pi}{-2}\right)}}} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{3}}{\frac{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{8} + {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right)}^{3}}\right), \color{blue}{\left(\frac{\frac{\mathsf{PI}\left(\right)}{\frac{4}{\mathsf{PI}\left(\right)}} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right)}^{2}}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{\frac{4}{\mathsf{PI}\left(\right)}} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right) \cdot \left(\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{x}{18 \cdot y}}}\right) + \frac{\mathsf{PI}\left(\right)}{-2}\right)}}\right)}\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} + {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{x}{18}}{y}}}\right)}^{3}} \cdot \left(\left(\frac{\pi}{\frac{4}{\pi}} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{x}{18}}{y}}}\right)}^{2}\right) \cdot \left(\frac{\pi}{\frac{4}{\pi}} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{x}{18}}{y}}}\right) \cdot \left(\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{x}{18}}{y}}}\right) + \frac{\pi}{-2}\right)\right)\right)} \]

9. Final simplification 100.0%

   \[\leadsto \frac{0.3333333333333333}{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} + {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{x}{18}}{y}}}\right)}^{3}} \cdot \left(\left(\frac{\pi}{\frac{4}{\pi}} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{x}{18}}{y}}}\right)}^{2}\right) \cdot \left(\frac{\pi}{\frac{4}{\pi}} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{x}{18}}{y}}}\right) \cdot \left(\frac{\pi}{-2} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{x}{18}}{y}}}\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(\frac{\sqrt{t}}{z \cdot \frac{18}{\frac{x}{y}}}\right)\\ \frac{0.3333333333333333 \cdot \left(\frac{\pi \cdot \pi}{4} - {t\_1}^{2}\right)}{t\_1 + \frac{\pi}{2}} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (asin (/ (sqrt t) (* z (/ 18.0 (/ x y)))))))
   (/
    (* 0.3333333333333333 (- (/ (* PI PI) 4.0) (pow t_1 2.0)))
    (+ t_1 (/ PI 2.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = asin((sqrt(t) / (z * (18.0 / (x / y)))));
	return (0.3333333333333333 * (((((double) M_PI) * ((double) M_PI)) / 4.0) - pow(t_1, 2.0))) / (t_1 + (((double) M_PI) / 2.0));
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.asin((Math.sqrt(t) / (z * (18.0 / (x / y)))));
	return (0.3333333333333333 * (((Math.PI * Math.PI) / 4.0) - Math.pow(t_1, 2.0))) / (t_1 + (Math.PI / 2.0));
}

Python

def code(x, y, z, t):
	t_1 = math.asin((math.sqrt(t) / (z * (18.0 / (x / y)))))
	return (0.3333333333333333 * (((math.pi * math.pi) / 4.0) - math.pow(t_1, 2.0))) / (t_1 + (math.pi / 2.0))

Julia

function code(x, y, z, t)
	t_1 = asin(Float64(sqrt(t) / Float64(z * Float64(18.0 / Float64(x / y)))))
	return Float64(Float64(0.3333333333333333 * Float64(Float64(Float64(pi * pi) / 4.0) - (t_1 ^ 2.0))) / Float64(t_1 + Float64(pi / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = asin((sqrt(t) / (z * (18.0 / (x / y)))));
	tmp = (0.3333333333333333 * (((pi * pi) / 4.0) - (t_1 ^ 2.0))) / (t_1 + (pi / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[ArcSin[N[(N[Sqrt[t], $MachinePrecision] / N[(z * N[(18.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(0.3333333333333333 * N[(N[(N[(Pi * Pi), $MachinePrecision] / 4.0), $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.5%

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. acos-asin N/A

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

   4. flip-- N/A

      \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \cdot \sin^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)}{\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)} \cdot \frac{1}{3} \]

   5. associate-*l/ N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \cdot \sin^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right) \cdot \frac{1}{3}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\left(\frac{\pi \cdot \pi}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{z \cdot \frac{18}{\frac{x}{y}}}\right)}^{2}\right) \cdot 0.3333333333333333}{\frac{\pi}{2} + \sin^{-1} \left(\frac{\sqrt{t}}{z \cdot \frac{18}{\frac{x}{y}}}\right)}} \]

5. Final simplification 98.5%

   \[\leadsto \frac{0.3333333333333333 \cdot \left(\frac{\pi \cdot \pi}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{z \cdot \frac{18}{\frac{x}{y}}}\right)}^{2}\right)}{\sin^{-1} \left(\frac{\sqrt{t}}{z \cdot \frac{18}{\frac{x}{y}}}\right) + \frac{\pi}{2}} \]
6. Add Preprocessing

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{z \cdot \frac{18}{\frac{x}{y}}}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* 0.3333333333333333 (acos (/ (sqrt t) (* z (/ 18.0 (/ x y)))))))

C

double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * acos((sqrt(t) / (z * (18.0 / (x / y)))));
}

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 = 0.3333333333333333d0 * acos((sqrt(t) / (z * (18.0d0 / (x / y)))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * Math.acos((Math.sqrt(t) / (z * (18.0 / (x / y)))));
}

Python

def code(x, y, z, t):
	return 0.3333333333333333 * math.acos((math.sqrt(t) / (z * (18.0 / (x / y)))))

Julia

function code(x, y, z, t)
	return Float64(0.3333333333333333 * acos(Float64(sqrt(t) / Float64(z * Float64(18.0 / Float64(x / y))))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.3333333333333333 * acos((sqrt(t) / (z * (18.0 / (x / y)))));
end

Wolfram

code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] / N[(z * N[(18.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right), \color{blue}{\frac{1}{3}}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\sqrt{t}}{z \cdot \frac{18}{\frac{x}{y}}}\right) \cdot 0.3333333333333333} \]

5. Final simplification 98.5%

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{z \cdot \frac{18}{\frac{x}{y}}}\right) \]
6. Add Preprocessing

Alternative 5: 97.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  0.3333333333333333
  (acos (* 0.05555555555555555 (/ (/ (* (sqrt t) x) z) y)))))

C

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

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 = 0.3333333333333333d0 * acos((0.05555555555555555d0 * (((sqrt(t) * x) / z) / y)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.3333333333333333 * acos((0.05555555555555555 * (((sqrt(t) * x) / z) / y)));
end

Wolfram

code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[ArcCos[N[(0.05555555555555555 * N[(N[(N[(N[Sqrt[t], $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{3}\right), \color{blue}{\cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \cos^{-1} \color{blue}{\left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)}\right) \]

   3. acos-lowering-acos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\left(3 \cdot \frac{x}{y \cdot 27}\right) \cdot \sqrt{t}}{z \cdot 2}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\left(3 \cdot \frac{x}{y \cdot 27}\right) \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3 \cdot x}{y \cdot 27} \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3 \cdot x}{27 \cdot y} \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\left(\frac{3}{27} \cdot \frac{x}{y}\right) \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3}{27} \cdot \left(\frac{x}{y} \cdot \sqrt{t}\right)}{2 \cdot z}\right)\right)\right) \]

   10. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3}{27}}{2} \cdot \frac{\frac{x}{y} \cdot \sqrt{t}}{z}\right)\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3}{27}}{2} \cdot \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{3}{27}}{2}\right), \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{9}}{2}\right), \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

   15. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{x \cdot \frac{\sqrt{t}}{z}}{y}\right)\right)\right)\right) \]

   16. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{\frac{x \cdot \sqrt{t}}{z}}{y}\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{\frac{\sqrt{t} \cdot x}{z}}{y}\right)\right)\right)\right) \]

   18. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{\sqrt{t} \cdot \frac{x}{z}}{y}\right)\right)\right)\right) \]

3. Simplified 97.3%

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

5. Final simplification 97.3%

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \frac{\frac{\sqrt{t} \cdot x}{z}}{y}\right) \]
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 2024149 
(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)))))