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

Percentage Accurate: 70.5% → 76.6%

Time: 17.4s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 70.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * Math.sqrt(x)) * Math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos y \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (cos y) (* 2.0 (sqrt x))) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (cos(y) * (2.0 * sqrt(x))) - (a / (b * 3.0));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (cos(y) * (2.0d0 * sqrt(x))) - (a / (b * 3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (Math.cos(y) * (2.0 * Math.sqrt(x))) - (a / (b * 3.0));
}

Python

def code(x, y, z, t, a, b):
	return (math.cos(y) * (2.0 * math.sqrt(x))) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(cos(y) * Float64(2.0 * sqrt(x))) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (cos(y) * (2.0 * sqrt(x))) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[Cos[y], $MachinePrecision] * N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

2. Taylor expanded in z around 0 80.6%

   \[\leadsto \color{blue}{2 \cdot \left(\cos y \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]
3. Step-by-step derivation

   1. *-commutative 80.6%

      \[\leadsto \color{blue}{\left(\cos y \cdot \sqrt{x}\right) \cdot 2} - \frac{a}{b \cdot 3} \]

   2. associate-*l* 80.6%

      \[\leadsto \color{blue}{\cos y \cdot \left(\sqrt{x} \cdot 2\right)} - \frac{a}{b \cdot 3} \]

   3. *-commutative 80.6%

      \[\leadsto \cos y \cdot \color{blue}{\left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

4. Simplified 80.6%

   \[\leadsto \color{blue}{\cos y \cdot \left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

5. Final simplification 80.6%

   \[\leadsto \cos y \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{b \cdot 3} \]

Alternative 2: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-69} \lor \neg \left(t_1 \leq 5 \cdot 10^{-104}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos y \cdot \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))))
   (if (or (<= t_1 -1e-69) (not (<= t_1 5e-104)))
     (- (* 2.0 (sqrt x)) t_1)
     (* 2.0 (* (cos y) (sqrt x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double tmp;
	if ((t_1 <= -1e-69) || !(t_1 <= 5e-104)) {
		tmp = (2.0 * sqrt(x)) - t_1;
	} else {
		tmp = 2.0 * (cos(y) * sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (b * 3.0d0)
    if ((t_1 <= (-1d-69)) .or. (.not. (t_1 <= 5d-104))) then
        tmp = (2.0d0 * sqrt(x)) - t_1
    else
        tmp = 2.0d0 * (cos(y) * sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double tmp;
	if ((t_1 <= -1e-69) || !(t_1 <= 5e-104)) {
		tmp = (2.0 * Math.sqrt(x)) - t_1;
	} else {
		tmp = 2.0 * (Math.cos(y) * Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	tmp = 0
	if (t_1 <= -1e-69) or not (t_1 <= 5e-104):
		tmp = (2.0 * math.sqrt(x)) - t_1
	else:
		tmp = 2.0 * (math.cos(y) * math.sqrt(x))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	tmp = 0.0
	if ((t_1 <= -1e-69) || !(t_1 <= 5e-104))
		tmp = Float64(Float64(2.0 * sqrt(x)) - t_1);
	else
		tmp = Float64(2.0 * Float64(cos(y) * sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (b * 3.0);
	tmp = 0.0;
	if ((t_1 <= -1e-69) || ~((t_1 <= 5e-104)))
		tmp = (2.0 * sqrt(x)) - t_1;
	else
		tmp = 2.0 * (cos(y) * sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-69], N[Not[LessEqual[t$95$1, 5e-104]], $MachinePrecision]], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(2.0 * N[(N[Cos[y], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b 3)) < -9.9999999999999996e-70 or 4.99999999999999979e-104 < (/.f64 a (*.f64 b 3))

   1. Initial program 85.9%

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

   2. Taylor expanded in z around 0 93.1%

      \[\leadsto \color{blue}{2 \cdot \left(\cos y \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]
   3. Step-by-step derivation

      1. *-commutative 93.1%

         \[\leadsto \color{blue}{\left(\cos y \cdot \sqrt{x}\right) \cdot 2} - \frac{a}{b \cdot 3} \]

      2. associate-*l* 93.1%

         \[\leadsto \color{blue}{\cos y \cdot \left(\sqrt{x} \cdot 2\right)} - \frac{a}{b \cdot 3} \]

      3. *-commutative 93.1%

         \[\leadsto \cos y \cdot \color{blue}{\left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

   4. Simplified 93.1%

      \[\leadsto \color{blue}{\cos y \cdot \left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

   5. Taylor expanded in y around 0 89.1%

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]

   if -9.9999999999999996e-70 < (/.f64 a (*.f64 b 3)) < 4.99999999999999979e-104

   1. Initial program 59.5%

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

   2. Taylor expanded in z around 0 60.7%

      \[\leadsto \color{blue}{2 \cdot \left(\cos y \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]
   3. Step-by-step derivation

      1. *-commutative 60.7%

         \[\leadsto \color{blue}{\left(\cos y \cdot \sqrt{x}\right) \cdot 2} - \frac{a}{b \cdot 3} \]

      2. associate-*l* 60.7%

         \[\leadsto \color{blue}{\cos y \cdot \left(\sqrt{x} \cdot 2\right)} - \frac{a}{b \cdot 3} \]

      3. *-commutative 60.7%

         \[\leadsto \cos y \cdot \color{blue}{\left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

   4. Simplified 60.7%

      \[\leadsto \color{blue}{\cos y \cdot \left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]
   5. Step-by-step derivation

      1. fma-neg 60.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, 2 \cdot \sqrt{x}, -\frac{a}{b \cdot 3}\right)} \]

   6. Applied egg-rr 60.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, 2 \cdot \sqrt{x}, -\frac{a}{b \cdot 3}\right)} \]

   7. Taylor expanded in a around 0 59.7%

      \[\leadsto \color{blue}{2 \cdot \left(\cos y \cdot \sqrt{x}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 59.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \cos y\right)} \]

   9. Simplified 59.7%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{b \cdot 3} \leq -1 \cdot 10^{-69} \lor \neg \left(\frac{a}{b \cdot 3} \leq 5 \cdot 10^{-104}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos y \cdot \sqrt{x}\right)\\ \end{array} \]

Alternative 3: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos y \cdot \left(2 \cdot \sqrt{x}\right) - a \cdot \frac{0.3333333333333333}{b} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (cos y) (* 2.0 (sqrt x))) (* a (/ 0.3333333333333333 b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (cos(y) * (2.0 * sqrt(x))) - (a * (0.3333333333333333 / b));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (cos(y) * (2.0d0 * sqrt(x))) - (a * (0.3333333333333333d0 / b))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (Math.cos(y) * (2.0 * Math.sqrt(x))) - (a * (0.3333333333333333 / b));
}

Python

def code(x, y, z, t, a, b):
	return (math.cos(y) * (2.0 * math.sqrt(x))) - (a * (0.3333333333333333 / b))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(cos(y) * Float64(2.0 * sqrt(x))) - Float64(a * Float64(0.3333333333333333 / b)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (cos(y) * (2.0 * sqrt(x))) - (a * (0.3333333333333333 / b));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[Cos[y], $MachinePrecision] * N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

2. Taylor expanded in z around 0 80.6%

   \[\leadsto \color{blue}{2 \cdot \left(\cos y \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]
3. Step-by-step derivation

   1. *-commutative 80.6%

      \[\leadsto \color{blue}{\left(\cos y \cdot \sqrt{x}\right) \cdot 2} - \frac{a}{b \cdot 3} \]

   2. associate-*l* 80.6%

      \[\leadsto \color{blue}{\cos y \cdot \left(\sqrt{x} \cdot 2\right)} - \frac{a}{b \cdot 3} \]

   3. *-commutative 80.6%

      \[\leadsto \cos y \cdot \color{blue}{\left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

4. Simplified 80.6%

   \[\leadsto \color{blue}{\cos y \cdot \left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

5. Taylor expanded in a around 0 80.5%

   \[\leadsto \cos y \cdot \left(2 \cdot \sqrt{x}\right) - \color{blue}{0.3333333333333333 \cdot \frac{a}{b}} \]
6. Step-by-step derivation

   1. *-commutative 80.5%

      \[\leadsto \cos y \cdot \left(2 \cdot \sqrt{x}\right) - \color{blue}{\frac{a}{b} \cdot 0.3333333333333333} \]

   2. metadata-eval 80.5%

      \[\leadsto \cos y \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{b} \cdot \color{blue}{\frac{1}{3}} \]

   3. times-frac 80.6%

      \[\leadsto \cos y \cdot \left(2 \cdot \sqrt{x}\right) - \color{blue}{\frac{a \cdot 1}{b \cdot 3}} \]

   4. associate-*r/ 80.5%

      \[\leadsto \cos y \cdot \left(2 \cdot \sqrt{x}\right) - \color{blue}{a \cdot \frac{1}{b \cdot 3}} \]

   5. *-commutative 80.5%

      \[\leadsto \cos y \cdot \left(2 \cdot \sqrt{x}\right) - a \cdot \frac{1}{\color{blue}{3 \cdot b}} \]

   6. associate-/r* 80.5%

      \[\leadsto \cos y \cdot \left(2 \cdot \sqrt{x}\right) - a \cdot \color{blue}{\frac{\frac{1}{3}}{b}} \]

   7. metadata-eval 80.5%

      \[\leadsto \cos y \cdot \left(2 \cdot \sqrt{x}\right) - a \cdot \frac{\color{blue}{0.3333333333333333}}{b} \]

7. Simplified 80.5%

   \[\leadsto \cos y \cdot \left(2 \cdot \sqrt{x}\right) - \color{blue}{a \cdot \frac{0.3333333333333333}{b}} \]

8. Final simplification 80.5%

   \[\leadsto \cos y \cdot \left(2 \cdot \sqrt{x}\right) - a \cdot \frac{0.3333333333333333}{b} \]

Alternative 4: 66.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{x} - \frac{a}{b \cdot 3} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (- (* 2.0 (sqrt x)) (/ a (* b 3.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (2.0 * sqrt(x)) - (a / (b * 3.0));
}

Fortran

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

Java

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

Python

def code(x, y, z, t, a, b):
	return (2.0 * math.sqrt(x)) - (a / (b * 3.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(2.0 * sqrt(x)) - Float64(a / Float64(b * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (2.0 * sqrt(x)) - (a / (b * 3.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

2. Taylor expanded in z around 0 80.6%

   \[\leadsto \color{blue}{2 \cdot \left(\cos y \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]
3. Step-by-step derivation

   1. *-commutative 80.6%

      \[\leadsto \color{blue}{\left(\cos y \cdot \sqrt{x}\right) \cdot 2} - \frac{a}{b \cdot 3} \]

   2. associate-*l* 80.6%

      \[\leadsto \color{blue}{\cos y \cdot \left(\sqrt{x} \cdot 2\right)} - \frac{a}{b \cdot 3} \]

   3. *-commutative 80.6%

      \[\leadsto \cos y \cdot \color{blue}{\left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

4. Simplified 80.6%

   \[\leadsto \color{blue}{\cos y \cdot \left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

5. Taylor expanded in y around 0 67.1%

   \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]

6. Final simplification 67.1%

   \[\leadsto 2 \cdot \sqrt{x} - \frac{a}{b \cdot 3} \]

Alternative 5: 51.2% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ a \cdot \frac{-0.3333333333333333}{b} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* a (/ -0.3333333333333333 b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return a * (-0.3333333333333333 / b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a * ((-0.3333333333333333d0) / b)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return a * (-0.3333333333333333 / b);
}

Python

def code(x, y, z, t, a, b):
	return a * (-0.3333333333333333 / b)

Julia

function code(x, y, z, t, a, b)
	return Float64(a * Float64(-0.3333333333333333 / b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = a * (-0.3333333333333333 / b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a * N[(-0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

2. Taylor expanded in z around 0 80.6%

   \[\leadsto \color{blue}{2 \cdot \left(\cos y \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]
3. Step-by-step derivation

   1. *-commutative 80.6%

      \[\leadsto \color{blue}{\left(\cos y \cdot \sqrt{x}\right) \cdot 2} - \frac{a}{b \cdot 3} \]

   2. associate-*l* 80.6%

      \[\leadsto \color{blue}{\cos y \cdot \left(\sqrt{x} \cdot 2\right)} - \frac{a}{b \cdot 3} \]

   3. *-commutative 80.6%

      \[\leadsto \cos y \cdot \color{blue}{\left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

4. Simplified 80.6%

   \[\leadsto \color{blue}{\cos y \cdot \left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]
5. Step-by-step derivation

   1. add-exp-log 60.0%

      \[\leadsto \color{blue}{e^{\log \left(\cos y \cdot \left(2 \cdot \sqrt{x}\right)\right)}} - \frac{a}{b \cdot 3} \]

6. Applied egg-rr 60.0%

   \[\leadsto \color{blue}{e^{\log \left(\cos y \cdot \left(2 \cdot \sqrt{x}\right)\right)}} - \frac{a}{b \cdot 3} \]
7. Step-by-step derivation

   1. add-exp-log 80.6%

      \[\leadsto \color{blue}{\cos y \cdot \left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

   2. associate-*r* 80.6%

      \[\leadsto \color{blue}{\left(\cos y \cdot 2\right) \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]

8. Applied egg-rr 80.6%

   \[\leadsto \color{blue}{\left(\cos y \cdot 2\right) \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]

9. Taylor expanded in x around 0 51.7%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
10. Step-by-step derivation

    1. associate-*r/ 51.8%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot a}{b}} \]

    2. *-commutative 51.8%

       \[\leadsto \frac{\color{blue}{a \cdot -0.3333333333333333}}{b} \]

    3. *-lft-identity 51.8%

       \[\leadsto \frac{a \cdot -0.3333333333333333}{\color{blue}{1 \cdot b}} \]

    4. times-frac 51.6%

       \[\leadsto \color{blue}{\frac{a}{1} \cdot \frac{-0.3333333333333333}{b}} \]

    5. /-rgt-identity 51.6%

       \[\leadsto \color{blue}{a} \cdot \frac{-0.3333333333333333}{b} \]

11. Simplified 51.6%

    \[\leadsto \color{blue}{a \cdot \frac{-0.3333333333333333}{b}} \]

12. Final simplification 51.6%

    \[\leadsto a \cdot \frac{-0.3333333333333333}{b} \]

Alternative 6: 51.2% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \frac{a}{b} \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (/ a b) -0.3333333333333333))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (a / b) * -0.3333333333333333;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a / b) * (-0.3333333333333333d0)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (a / b) * -0.3333333333333333;
}

Python

def code(x, y, z, t, a, b):
	return (a / b) * -0.3333333333333333

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(a / b) * -0.3333333333333333)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (a / b) * -0.3333333333333333;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a / b), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

2. Taylor expanded in z around 0 80.6%

   \[\leadsto \color{blue}{2 \cdot \left(\cos y \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]
3. Step-by-step derivation

   1. *-commutative 80.6%

      \[\leadsto \color{blue}{\left(\cos y \cdot \sqrt{x}\right) \cdot 2} - \frac{a}{b \cdot 3} \]

   2. associate-*l* 80.6%

      \[\leadsto \color{blue}{\cos y \cdot \left(\sqrt{x} \cdot 2\right)} - \frac{a}{b \cdot 3} \]

   3. *-commutative 80.6%

      \[\leadsto \cos y \cdot \color{blue}{\left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

4. Simplified 80.6%

   \[\leadsto \color{blue}{\cos y \cdot \left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

5. Taylor expanded in x around 0 51.7%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
6. Step-by-step derivation

   1. *-commutative 51.7%

      \[\leadsto \color{blue}{\frac{a}{b} \cdot -0.3333333333333333} \]

7. Simplified 51.7%

   \[\leadsto \color{blue}{\frac{a}{b} \cdot -0.3333333333333333} \]

8. Final simplification 51.7%

   \[\leadsto \frac{a}{b} \cdot -0.3333333333333333 \]

Alternative 7: 51.3% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \frac{a}{b \cdot -3} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ a (* b -3.0)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return a / (b * -3.0);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a / (b * (-3.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return a / (b * -3.0);
}

Python

def code(x, y, z, t, a, b):
	return a / (b * -3.0)

Julia

function code(x, y, z, t, a, b)
	return Float64(a / Float64(b * -3.0))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = a / (b * -3.0);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

2. Taylor expanded in z around 0 80.6%

   \[\leadsto \color{blue}{2 \cdot \left(\cos y \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]
3. Step-by-step derivation

   1. *-commutative 80.6%

      \[\leadsto \color{blue}{\left(\cos y \cdot \sqrt{x}\right) \cdot 2} - \frac{a}{b \cdot 3} \]

   2. associate-*l* 80.6%

      \[\leadsto \color{blue}{\cos y \cdot \left(\sqrt{x} \cdot 2\right)} - \frac{a}{b \cdot 3} \]

   3. *-commutative 80.6%

      \[\leadsto \cos y \cdot \color{blue}{\left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]

4. Simplified 80.6%

   \[\leadsto \color{blue}{\cos y \cdot \left(2 \cdot \sqrt{x}\right)} - \frac{a}{b \cdot 3} \]
5. Step-by-step derivation

   1. fma-neg 80.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, 2 \cdot \sqrt{x}, -\frac{a}{b \cdot 3}\right)} \]

6. Applied egg-rr 80.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, 2 \cdot \sqrt{x}, -\frac{a}{b \cdot 3}\right)} \]

7. Taylor expanded in x around 0 51.7%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
8. Step-by-step derivation

   1. *-commutative 51.7%

      \[\leadsto \color{blue}{\frac{a}{b} \cdot -0.3333333333333333} \]

   2. associate-/r/ 51.7%

      \[\leadsto \color{blue}{\frac{a}{\frac{b}{-0.3333333333333333}}} \]

9. Simplified 51.7%

   \[\leadsto \color{blue}{\frac{a}{\frac{b}{-0.3333333333333333}}} \]

10. Taylor expanded in b around 0 51.8%

    \[\leadsto \frac{a}{\color{blue}{-3 \cdot b}} \]
11. Step-by-step derivation

    1. *-commutative 51.8%

       \[\leadsto \frac{a}{\color{blue}{b \cdot -3}} \]

12. Simplified 51.8%

    \[\leadsto \frac{a}{\color{blue}{b \cdot -3}} \]

13. Final simplification 51.8%

    \[\leadsto \frac{a}{b \cdot -3} \]

Developer target: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{0.3333333333333333}{z}}{t}\\ t_2 := \frac{\frac{a}{3}}{b}\\ t_3 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;t_3 \cdot \cos \left(\frac{1}{y} - t_1\right) - t_2\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - t_1\right) \cdot t_3 - \frac{\frac{a}{b}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (/ 0.3333333333333333 z) t))
        (t_2 (/ (/ a 3.0) b))
        (t_3 (* 2.0 (sqrt x))))
   (if (< z -1.3793337487235141e+129)
     (- (* t_3 (cos (- (/ 1.0 y) t_1))) t_2)
     (if (< z 3.516290613555987e+106)
       (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) t_2)
       (- (* (cos (- y t_1)) t_3) (/ (/ a b) 3.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (0.3333333333333333d0 / z) / t
    t_2 = (a / 3.0d0) / b
    t_3 = 2.0d0 * sqrt(x)
    if (z < (-1.3793337487235141d+129)) then
        tmp = (t_3 * cos(((1.0d0 / y) - t_1))) - t_2
    else if (z < 3.516290613555987d+106) then
        tmp = ((sqrt(x) * 2.0d0) * cos((y - ((t / 3.0d0) * z)))) - t_2
    else
        tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * Math.sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * Math.cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((Math.sqrt(x) * 2.0) * Math.cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (Math.cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (0.3333333333333333 / z) / t
	t_2 = (a / 3.0) / b
	t_3 = 2.0 * math.sqrt(x)
	tmp = 0
	if z < -1.3793337487235141e+129:
		tmp = (t_3 * math.cos(((1.0 / y) - t_1))) - t_2
	elif z < 3.516290613555987e+106:
		tmp = ((math.sqrt(x) * 2.0) * math.cos((y - ((t / 3.0) * z)))) - t_2
	else:
		tmp = (math.cos((y - t_1)) * t_3) - ((a / b) / 3.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(0.3333333333333333 / z) / t)
	t_2 = Float64(Float64(a / 3.0) / b)
	t_3 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (z < -1.3793337487235141e+129)
		tmp = Float64(Float64(t_3 * cos(Float64(Float64(1.0 / y) - t_1))) - t_2);
	elseif (z < 3.516290613555987e+106)
		tmp = Float64(Float64(Float64(sqrt(x) * 2.0) * cos(Float64(y - Float64(Float64(t / 3.0) * z)))) - t_2);
	else
		tmp = Float64(Float64(cos(Float64(y - t_1)) * t_3) - Float64(Float64(a / b) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (0.3333333333333333 / z) / t;
	t_2 = (a / 3.0) / b;
	t_3 = 2.0 * sqrt(x);
	tmp = 0.0;
	if (z < -1.3793337487235141e+129)
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	elseif (z < 3.516290613555987e+106)
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	else
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.3793337487235141e+129], N[(N[(t$95$3 * N[Cos[N[(N[(1.0 / y), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[z, 3.516290613555987e+106], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(y - N[(N[(t / 3.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(N[Cos[N[(y - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2023196 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1.0 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))