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

Percentage Accurate: 98.0% → 98.0%

Time: 5.0s

Alternatives: 7

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

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: 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]

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{3} \cdot \frac{1}{3}\\ \mathbf{if}\;\sqrt{t} \leq 0.62:\\ \;\;\;\;\frac{1}{3} \cdot \cos^{-1} \left(t\_1 \cdot \sqrt{t}\right)\\ \mathbf{elif}\;\sqrt{t} \leq 7.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{3} \cdot \cos^{-1} \left(y \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3} \cdot \cos^{-1} t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ 1.0 3.0) (/ 1.0 3.0))))
   (if (<= (sqrt t) 0.62)
     (* (/ 1.0 3.0) (acos (* t_1 (sqrt t))))
     (if (<= (sqrt t) 7.5e+58)
       (* (/ 1.0 3.0) (acos (* y 27.0)))
       (* (/ 1.0 3.0) (acos t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (1.0 / 3.0) * (1.0 / 3.0);
	double tmp;
	if (sqrt(t) <= 0.62) {
		tmp = (1.0 / 3.0) * acos((t_1 * sqrt(t)));
	} else if (sqrt(t) <= 7.5e+58) {
		tmp = (1.0 / 3.0) * acos((y * 27.0));
	} else {
		tmp = (1.0 / 3.0) * acos(t_1);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (1.0d0 / 3.0d0) * (1.0d0 / 3.0d0)
    if (sqrt(t) <= 0.62d0) then
        tmp = (1.0d0 / 3.0d0) * acos((t_1 * sqrt(t)))
    else if (sqrt(t) <= 7.5d+58) then
        tmp = (1.0d0 / 3.0d0) * acos((y * 27.0d0))
    else
        tmp = (1.0d0 / 3.0d0) * acos(t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (1.0 / 3.0) * (1.0 / 3.0);
	double tmp;
	if (Math.sqrt(t) <= 0.62) {
		tmp = (1.0 / 3.0) * Math.acos((t_1 * Math.sqrt(t)));
	} else if (Math.sqrt(t) <= 7.5e+58) {
		tmp = (1.0 / 3.0) * Math.acos((y * 27.0));
	} else {
		tmp = (1.0 / 3.0) * Math.acos(t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (1.0 / 3.0) * (1.0 / 3.0)
	tmp = 0
	if math.sqrt(t) <= 0.62:
		tmp = (1.0 / 3.0) * math.acos((t_1 * math.sqrt(t)))
	elif math.sqrt(t) <= 7.5e+58:
		tmp = (1.0 / 3.0) * math.acos((y * 27.0))
	else:
		tmp = (1.0 / 3.0) * math.acos(t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(1.0 / 3.0) * Float64(1.0 / 3.0))
	tmp = 0.0
	if (sqrt(t) <= 0.62)
		tmp = Float64(Float64(1.0 / 3.0) * acos(Float64(t_1 * sqrt(t))));
	elseif (sqrt(t) <= 7.5e+58)
		tmp = Float64(Float64(1.0 / 3.0) * acos(Float64(y * 27.0)));
	else
		tmp = Float64(Float64(1.0 / 3.0) * acos(t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (1.0 / 3.0) * (1.0 / 3.0);
	tmp = 0.0;
	if (sqrt(t) <= 0.62)
		tmp = (1.0 / 3.0) * acos((t_1 * sqrt(t)));
	elseif (sqrt(t) <= 7.5e+58)
		tmp = (1.0 / 3.0) * acos((y * 27.0));
	else
		tmp = (1.0 / 3.0) * acos(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(1.0 / 3.0), $MachinePrecision] * N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[t], $MachinePrecision], 0.62], N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(t$95$1 * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sqrt[t], $MachinePrecision], 7.5e+58], N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(y * 27.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[t$95$1], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 t) < 0.619999999999999996

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{1}{9} \cdot \frac{x}{y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\left(\frac{1}{18} \cdot \frac{x}{y \cdot z}\right)} \cdot \sqrt{t}\right) \]

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\left(\frac{1}{18} \cdot \frac{x}{y \cdot z}\right)} \cdot \sqrt{t}\right) \]

   11. Applied rewrites 47.0%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\left(\frac{1}{18} \cdot \frac{x}{y \cdot z}\right)} \cdot \sqrt{t}\right) \]

   14. Applied rewrites 93.8%

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

   if 0.619999999999999996 < (sqrt.f64 t) < 7.5000000000000001e58

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{1}{9} \cdot \frac{x}{y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   8. Applied rewrites 23.9%

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{3}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   11. Applied rewrites 34.2%

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

   if 7.5000000000000001e58 < (sqrt.f64 t)

   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. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{1}{9} \cdot \frac{x}{y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   8. Applied rewrites 23.9%

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{3}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{t} \leq 0.28:\\ \;\;\;\;\frac{1}{3} \cdot \cos^{-1} \left(\frac{1}{3} \cdot \sqrt{t}\right)\\ \mathbf{elif}\;\sqrt{t} \leq 7.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{3} \cdot \cos^{-1} \left(y \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3} \cdot \cos^{-1} \left(\frac{1}{3} \cdot \frac{1}{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (sqrt t) 0.28)
   (* (/ 1.0 3.0) (acos (* (/ 1.0 3.0) (sqrt t))))
   (if (<= (sqrt t) 7.5e+58)
     (* (/ 1.0 3.0) (acos (* y 27.0)))
     (* (/ 1.0 3.0) (acos (* (/ 1.0 3.0) (/ 1.0 3.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (sqrt(t) <= 0.28) {
		tmp = (1.0 / 3.0) * acos(((1.0 / 3.0) * sqrt(t)));
	} else if (sqrt(t) <= 7.5e+58) {
		tmp = (1.0 / 3.0) * acos((y * 27.0));
	} else {
		tmp = (1.0 / 3.0) * acos(((1.0 / 3.0) * (1.0 / 3.0)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (sqrt(t) <= 0.28d0) then
        tmp = (1.0d0 / 3.0d0) * acos(((1.0d0 / 3.0d0) * sqrt(t)))
    else if (sqrt(t) <= 7.5d+58) then
        tmp = (1.0d0 / 3.0d0) * acos((y * 27.0d0))
    else
        tmp = (1.0d0 / 3.0d0) * acos(((1.0d0 / 3.0d0) * (1.0d0 / 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.sqrt(t) <= 0.28) {
		tmp = (1.0 / 3.0) * Math.acos(((1.0 / 3.0) * Math.sqrt(t)));
	} else if (Math.sqrt(t) <= 7.5e+58) {
		tmp = (1.0 / 3.0) * Math.acos((y * 27.0));
	} else {
		tmp = (1.0 / 3.0) * Math.acos(((1.0 / 3.0) * (1.0 / 3.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.sqrt(t) <= 0.28:
		tmp = (1.0 / 3.0) * math.acos(((1.0 / 3.0) * math.sqrt(t)))
	elif math.sqrt(t) <= 7.5e+58:
		tmp = (1.0 / 3.0) * math.acos((y * 27.0))
	else:
		tmp = (1.0 / 3.0) * math.acos(((1.0 / 3.0) * (1.0 / 3.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (sqrt(t) <= 0.28)
		tmp = Float64(Float64(1.0 / 3.0) * acos(Float64(Float64(1.0 / 3.0) * sqrt(t))));
	elseif (sqrt(t) <= 7.5e+58)
		tmp = Float64(Float64(1.0 / 3.0) * acos(Float64(y * 27.0)));
	else
		tmp = Float64(Float64(1.0 / 3.0) * acos(Float64(Float64(1.0 / 3.0) * Float64(1.0 / 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (sqrt(t) <= 0.28)
		tmp = (1.0 / 3.0) * acos(((1.0 / 3.0) * sqrt(t)));
	elseif (sqrt(t) <= 7.5e+58)
		tmp = (1.0 / 3.0) * acos((y * 27.0));
	else
		tmp = (1.0 / 3.0) * acos(((1.0 / 3.0) * (1.0 / 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Sqrt[t], $MachinePrecision], 0.28], N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(N[(1.0 / 3.0), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sqrt[t], $MachinePrecision], 7.5e+58], N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(y * 27.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(N[(1.0 / 3.0), $MachinePrecision] * N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 t) < 0.28000000000000003

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{1}{9} \cdot \frac{x}{y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\left(\frac{1}{18} \cdot \frac{x}{y \cdot z}\right)} \cdot \sqrt{t}\right) \]

   8. Applied rewrites 93.6%

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

   if 0.28000000000000003 < (sqrt.f64 t) < 7.5000000000000001e58

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{1}{9} \cdot \frac{x}{y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   8. Applied rewrites 23.9%

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{3}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   11. Applied rewrites 34.2%

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

   if 7.5000000000000001e58 < (sqrt.f64 t)

   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. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{1}{9} \cdot \frac{x}{y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   8. Applied rewrites 23.9%

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{3}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 41.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot 27 \leq -3.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{1}{3}\\ \mathbf{elif}\;y \cdot 27 \leq 2.3 \cdot 10^{+75}:\\ \;\;\;\;\frac{1}{3} \cdot \cos^{-1} \left(\left(y \cdot 27\right) \cdot \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y 27.0) -3.5e+91)
   (/ 1.0 3.0)
   (if (<= (* y 27.0) 2.3e+75)
     (* (/ 1.0 3.0) (acos (* (* y 27.0) (sqrt t))))
     (/ 1.0 3.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * 27.0) <= -3.5e+91) {
		tmp = 1.0 / 3.0;
	} else if ((y * 27.0) <= 2.3e+75) {
		tmp = (1.0 / 3.0) * acos(((y * 27.0) * sqrt(t)));
	} else {
		tmp = 1.0 / 3.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y * 27.0d0) <= (-3.5d+91)) then
        tmp = 1.0d0 / 3.0d0
    else if ((y * 27.0d0) <= 2.3d+75) then
        tmp = (1.0d0 / 3.0d0) * acos(((y * 27.0d0) * sqrt(t)))
    else
        tmp = 1.0d0 / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * 27.0) <= -3.5e+91) {
		tmp = 1.0 / 3.0;
	} else if ((y * 27.0) <= 2.3e+75) {
		tmp = (1.0 / 3.0) * Math.acos(((y * 27.0) * Math.sqrt(t)));
	} else {
		tmp = 1.0 / 3.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y * 27.0) <= -3.5e+91:
		tmp = 1.0 / 3.0
	elif (y * 27.0) <= 2.3e+75:
		tmp = (1.0 / 3.0) * math.acos(((y * 27.0) * math.sqrt(t)))
	else:
		tmp = 1.0 / 3.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * 27.0) <= -3.5e+91)
		tmp = Float64(1.0 / 3.0);
	elseif (Float64(y * 27.0) <= 2.3e+75)
		tmp = Float64(Float64(1.0 / 3.0) * acos(Float64(Float64(y * 27.0) * sqrt(t))));
	else
		tmp = Float64(1.0 / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y * 27.0) <= -3.5e+91)
		tmp = 1.0 / 3.0;
	elseif ((y * 27.0) <= 2.3e+75)
		tmp = (1.0 / 3.0) * acos(((y * 27.0) * sqrt(t)));
	else
		tmp = 1.0 / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y * 27.0), $MachinePrecision], -3.5e+91], N[(1.0 / 3.0), $MachinePrecision], If[LessEqual[N[(y * 27.0), $MachinePrecision], 2.3e+75], N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(N[(y * 27.0), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 27 binary64)) < -3.50000000000000001e91 or 2.2999999999999999e75 < (*.f64 y #s(literal 27 binary64))

   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. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   5. Applied rewrites 16.9%

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{1}{3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   8. Applied rewrites 2.7%

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\left(y \cdot 27\right)} \]

   9. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y \cdot z}\right)\right)} \]

   11. Applied rewrites 18.0%

       \[\leadsto \color{blue}{\cos^{-1} \left(\frac{1}{3} \cdot \frac{1}{3}\right)} \]

   12. Taylor expanded in t around -inf

       \[\leadsto \cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y \cdot z}\right)\right) \]

   14. Applied rewrites 19.5%

       \[\leadsto \frac{1}{\color{blue}{3}} \]

   if -3.50000000000000001e91 < (*.f64 y #s(literal 27 binary64)) < 2.2999999999999999e75

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{1}{9} \cdot \frac{x}{y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\left(\frac{1}{18} \cdot \frac{x}{y \cdot z}\right)} \cdot \sqrt{t}\right) \]

   8. Applied rewrites 61.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\left(\frac{1}{18} \cdot \frac{x}{y \cdot z}\right)} \cdot \sqrt{t}\right) \]

   11. Applied rewrites 67.4%

       \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\left(y \cdot 27\right)} \cdot \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 65.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{t} \leq 0.28:\\ \;\;\;\;\frac{1}{3} \cdot \cos^{-1} \left(\frac{1}{3} \cdot \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3} \cdot \cos^{-1} \left(y \cdot 27\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (sqrt t) 0.28)
   (* (/ 1.0 3.0) (acos (* (/ 1.0 3.0) (sqrt t))))
   (* (/ 1.0 3.0) (acos (* y 27.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (sqrt(t) <= 0.28) {
		tmp = (1.0 / 3.0) * acos(((1.0 / 3.0) * sqrt(t)));
	} else {
		tmp = (1.0 / 3.0) * acos((y * 27.0));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (sqrt(t) <= 0.28d0) then
        tmp = (1.0d0 / 3.0d0) * acos(((1.0d0 / 3.0d0) * sqrt(t)))
    else
        tmp = (1.0d0 / 3.0d0) * acos((y * 27.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.sqrt(t) <= 0.28) {
		tmp = (1.0 / 3.0) * Math.acos(((1.0 / 3.0) * Math.sqrt(t)));
	} else {
		tmp = (1.0 / 3.0) * Math.acos((y * 27.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.sqrt(t) <= 0.28:
		tmp = (1.0 / 3.0) * math.acos(((1.0 / 3.0) * math.sqrt(t)))
	else:
		tmp = (1.0 / 3.0) * math.acos((y * 27.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (sqrt(t) <= 0.28)
		tmp = Float64(Float64(1.0 / 3.0) * acos(Float64(Float64(1.0 / 3.0) * sqrt(t))));
	else
		tmp = Float64(Float64(1.0 / 3.0) * acos(Float64(y * 27.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (sqrt(t) <= 0.28)
		tmp = (1.0 / 3.0) * acos(((1.0 / 3.0) * sqrt(t)));
	else
		tmp = (1.0 / 3.0) * acos((y * 27.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Sqrt[t], $MachinePrecision], 0.28], N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(N[(1.0 / 3.0), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(y * 27.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 t) < 0.28000000000000003

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{1}{9} \cdot \frac{x}{y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\left(\frac{1}{18} \cdot \frac{x}{y \cdot z}\right)} \cdot \sqrt{t}\right) \]

   8. Applied rewrites 93.6%

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

   if 0.28000000000000003 < (sqrt.f64 t)

   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. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{1}{9} \cdot \frac{x}{y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   8. Applied rewrites 23.9%

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{3}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   11. Applied rewrites 21.9%

       \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(y \cdot 27\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 41.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot 27 \leq -0.88:\\ \;\;\;\;\frac{1}{3}\\ \mathbf{elif}\;y \cdot 27 \leq 0.52:\\ \;\;\;\;\frac{1}{3} \cdot \cos^{-1} \left(y \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y 27.0) -0.88)
   (/ 1.0 3.0)
   (if (<= (* y 27.0) 0.52) (* (/ 1.0 3.0) (acos (* y 27.0))) (/ 1.0 3.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * 27.0) <= -0.88) {
		tmp = 1.0 / 3.0;
	} else if ((y * 27.0) <= 0.52) {
		tmp = (1.0 / 3.0) * acos((y * 27.0));
	} else {
		tmp = 1.0 / 3.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y * 27.0d0) <= (-0.88d0)) then
        tmp = 1.0d0 / 3.0d0
    else if ((y * 27.0d0) <= 0.52d0) then
        tmp = (1.0d0 / 3.0d0) * acos((y * 27.0d0))
    else
        tmp = 1.0d0 / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * 27.0) <= -0.88) {
		tmp = 1.0 / 3.0;
	} else if ((y * 27.0) <= 0.52) {
		tmp = (1.0 / 3.0) * Math.acos((y * 27.0));
	} else {
		tmp = 1.0 / 3.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y * 27.0) <= -0.88:
		tmp = 1.0 / 3.0
	elif (y * 27.0) <= 0.52:
		tmp = (1.0 / 3.0) * math.acos((y * 27.0))
	else:
		tmp = 1.0 / 3.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * 27.0) <= -0.88)
		tmp = Float64(1.0 / 3.0);
	elseif (Float64(y * 27.0) <= 0.52)
		tmp = Float64(Float64(1.0 / 3.0) * acos(Float64(y * 27.0)));
	else
		tmp = Float64(1.0 / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y * 27.0) <= -0.88)
		tmp = 1.0 / 3.0;
	elseif ((y * 27.0) <= 0.52)
		tmp = (1.0 / 3.0) * acos((y * 27.0));
	else
		tmp = 1.0 / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y * 27.0), $MachinePrecision], -0.88], N[(1.0 / 3.0), $MachinePrecision], If[LessEqual[N[(y * 27.0), $MachinePrecision], 0.52], N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(y * 27.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y #s(literal 27 binary64)) < -0.880000000000000004 or 0.52000000000000002 < (*.f64 y #s(literal 27 binary64))

   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. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   5. Applied rewrites 16.9%

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{1}{3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   8. Applied rewrites 3.4%

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\left(y \cdot 27\right)} \]

   9. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y \cdot z}\right)\right)} \]

   11. Applied rewrites 18.0%

       \[\leadsto \color{blue}{\cos^{-1} \left(\frac{1}{3} \cdot \frac{1}{3}\right)} \]

   12. Taylor expanded in t around -inf

       \[\leadsto \cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y \cdot z}\right)\right) \]

   14. Applied rewrites 19.5%

       \[\leadsto \frac{1}{\color{blue}{3}} \]

   if -0.880000000000000004 < (*.f64 y #s(literal 27 binary64)) < 0.52000000000000002

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{1}{9} \cdot \frac{x}{y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   8. Applied rewrites 23.9%

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{3}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

   11. Applied rewrites 89.8%

       \[\leadsto \frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(y \cdot 27\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 19.5% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{3} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ 1.0 3.0))

C

double code(double x, double y, double z, double t) {
	return 1.0 / 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 = 1.0d0 / 3.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 / 3.0;
}

Python

def code(x, y, z, t):
	return 1.0 / 3.0

Julia

function code(x, y, z, t)
	return Float64(1.0 / 3.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 / 3.0;
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 / 3.0), $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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \cdot \color{blue}{\cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

5. Applied rewrites 16.9%

   \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{1}{3}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \cdot \color{blue}{\cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)} \]

8. Applied rewrites 3.7%

   \[\leadsto \frac{1}{3} \cdot \color{blue}{\left(y \cdot 27\right)} \]

9. Taylor expanded in t around -inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y \cdot z}\right)\right)} \]

11. Applied rewrites 18.0%

    \[\leadsto \color{blue}{\cos^{-1} \left(\frac{1}{3} \cdot \frac{1}{3}\right)} \]

12. Taylor expanded in t around -inf

    \[\leadsto \cos^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y \cdot z}\right)\right) \]

14. Applied rewrites 19.5%

    \[\leadsto \frac{1}{\color{blue}{3}} \]
15. Add Preprocessing

Developer Target 1: 98.1% 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 2024321 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64
  :pre (TRUE)

  :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)))))