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

Percentage Accurate: 70.9% → 77.2%

Time: 18.7s

Alternatives: 8

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: 70.9% 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: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\sqrt{x} \cdot \cos y\right) + \frac{a}{3} \cdot \frac{-1}{b} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (* 2.0 (* (sqrt x) (cos y))) (* (/ a 3.0) (/ -1.0 b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (2.0 * (sqrt(x) * cos(y))) + ((a / 3.0) * (-1.0 / 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 = (2.0d0 * (sqrt(x) * cos(y))) + ((a / 3.0d0) * ((-1.0d0) / b))
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))) + ((a / 3.0) * (-1.0 / b));
}

Python

def code(x, y, z, t, a, b):
	return (2.0 * (math.sqrt(x) * math.cos(y))) + ((a / 3.0) * (-1.0 / b))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(2.0 * Float64(sqrt(x) * cos(y))) + Float64(Float64(a / 3.0) * Float64(-1.0 / b)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (2.0 * (sqrt(x) * cos(y))) + ((a / 3.0) * (-1.0 / b));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a / 3.0), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.0%

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

   1. associate-*l* 72.0%

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

   2. fma-neg 72.0%

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

   3. remove-double-neg 72.0%

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

   4. fma-neg 72.0%

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

   5. remove-double-neg 72.0%

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

   6. associate-/l* 71.8%

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

   7. *-commutative 71.8%

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

3. Simplified 71.8%

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

   1. *-un-lft-identity 71.8%

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

   2. *-commutative 71.8%

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

   3. times-frac 71.8%

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

5. Applied egg-rr 71.8%

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

6. Taylor expanded in z around 0 76.9%

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

7. Final simplification 76.9%

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

Alternative 2: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-70} \lor \neg \left(t_1 \leq 5 \cdot 10^{-111}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))))
   (if (or (<= t_1 -5e-70) (not (<= t_1 5e-111)))
     (- (* 2.0 (sqrt x)) t_1)
     (* 2.0 (* (sqrt x) (cos y))))))

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 <= -5e-70) || !(t_1 <= 5e-111)) {
		tmp = (2.0 * sqrt(x)) - t_1;
	} else {
		tmp = 2.0 * (sqrt(x) * cos(y));
	}
	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 <= (-5d-70)) .or. (.not. (t_1 <= 5d-111))) then
        tmp = (2.0d0 * sqrt(x)) - t_1
    else
        tmp = 2.0d0 * (sqrt(x) * cos(y))
    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 <= -5e-70) || !(t_1 <= 5e-111)) {
		tmp = (2.0 * Math.sqrt(x)) - t_1;
	} else {
		tmp = 2.0 * (Math.sqrt(x) * Math.cos(y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (b * 3.0)
	tmp = 0
	if (t_1 <= -5e-70) or not (t_1 <= 5e-111):
		tmp = (2.0 * math.sqrt(x)) - t_1
	else:
		tmp = 2.0 * (math.sqrt(x) * math.cos(y))
	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 <= -5e-70) || !(t_1 <= 5e-111))
		tmp = Float64(Float64(2.0 * sqrt(x)) - t_1);
	else
		tmp = Float64(2.0 * Float64(sqrt(x) * cos(y)));
	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 <= -5e-70) || ~((t_1 <= 5e-111)))
		tmp = (2.0 * sqrt(x)) - t_1;
	else
		tmp = 2.0 * (sqrt(x) * cos(y));
	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, -5e-70], N[Not[LessEqual[t$95$1, 5e-111]], $MachinePrecision]], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b 3)) < -4.9999999999999998e-70 or 5.0000000000000003e-111 < (/.f64 a (*.f64 b 3))

   1. Initial program 82.5%

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

   2. Taylor expanded in z around 0 90.7%

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

      1. *-commutative 90.7%

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

      2. associate-*l* 90.7%

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

      3. *-commutative 90.7%

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

   4. Simplified 90.7%

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

   5. Taylor expanded in y around 0 85.8%

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

   if -4.9999999999999998e-70 < (/.f64 a (*.f64 b 3)) < 5.0000000000000003e-111

   1. Initial program 56.6%

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

   2. Taylor expanded in z around 0 56.7%

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

      1. *-commutative 56.7%

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

      2. associate-*l* 56.7%

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

      3. *-commutative 56.7%

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

   4. Simplified 56.7%

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

      1. associate-*r* 56.7%

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

      2. fma-neg 56.7%

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

   6. Applied egg-rr 56.7%

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

   7. Taylor expanded in a around 0 55.5%

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

      1. *-commutative 55.5%

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

   9. Simplified 55.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{b \cdot 3} \leq -5 \cdot 10^{-70} \lor \neg \left(\frac{a}{b \cdot 3} \leq 5 \cdot 10^{-111}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos y\right)\\ \end{array} \]

Alternative 3: 77.2% 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 72.0%

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

2. Taylor expanded in z around 0 76.9%

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

   1. *-commutative 76.9%

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

   2. associate-*l* 76.9%

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

   3. *-commutative 76.9%

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

4. Simplified 76.9%

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

5. Final simplification 76.9%

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

Alternative 4: 66.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{x} - 0.3333333333333333 \cdot \frac{a}{b} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (2.0 * sqrt(x)) - (0.3333333333333333 * (a / 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 = (2.0d0 * sqrt(x)) - (0.3333333333333333d0 * (a / b))
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)) - (0.3333333333333333 * (a / b));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

2. Taylor expanded in z around 0 76.9%

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

   1. *-commutative 76.9%

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

   2. associate-*l* 76.9%

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

   3. *-commutative 76.9%

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

4. Simplified 76.9%

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

5. Taylor expanded in y around 0 66.3%

   \[\leadsto \color{blue}{2 \cdot \sqrt{x} - 0.3333333333333333 \cdot \frac{a}{b}} \]

6. Final simplification 66.3%

   \[\leadsto 2 \cdot \sqrt{x} - 0.3333333333333333 \cdot \frac{a}{b} \]

Alternative 5: 66.2% 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 72.0%

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

2. Taylor expanded in z around 0 76.9%

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

   1. *-commutative 76.9%

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

   2. associate-*l* 76.9%

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

   3. *-commutative 76.9%

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

4. Simplified 76.9%

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

5. Taylor expanded in y around 0 66.4%

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

6. Final simplification 66.4%

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

Alternative 6: 51.1% 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 72.0%

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

2. Taylor expanded in z around 0 76.9%

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

   1. *-commutative 76.9%

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

   2. associate-*l* 76.9%

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

   3. *-commutative 76.9%

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

4. Simplified 76.9%

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

5. Taylor expanded in x around 0 46.5%

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

6. Final simplification 46.5%

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

Alternative 7: 51.1% accurate, 43.4× speedup

Math

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

FPCore

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

C

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

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 = (-0.3333333333333333d0) / (b / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

2. Taylor expanded in z around 0 76.9%

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

   1. *-commutative 76.9%

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

   2. associate-*l* 76.9%

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

   3. *-commutative 76.9%

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

4. Simplified 76.9%

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

5. Taylor expanded in x around 0 46.5%

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

   1. associate-*r/ 46.5%

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

   2. *-commutative 46.5%

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

   3. *-rgt-identity 46.5%

      \[\leadsto \frac{\color{blue}{\left(a \cdot -0.3333333333333333\right) \cdot 1}}{b} \]

   4. associate-*r/ 46.5%

      \[\leadsto \color{blue}{\left(a \cdot -0.3333333333333333\right) \cdot \frac{1}{b}} \]

   5. associate-*l* 46.5%

      \[\leadsto \color{blue}{a \cdot \left(-0.3333333333333333 \cdot \frac{1}{b}\right)} \]

   6. associate-*r/ 46.5%

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

   7. metadata-eval 46.5%

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

7. Simplified 46.5%

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

8. Taylor expanded in a around 0 46.5%

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

   1. associate-*r/ 46.5%

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

   2. associate-/l* 46.6%

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

10. Simplified 46.6%

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

11. Final simplification 46.6%

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

Alternative 8: 51.2% 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 72.0%

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

2. Taylor expanded in z around 0 76.9%

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

   1. *-commutative 76.9%

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

   2. associate-*l* 76.9%

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

   3. *-commutative 76.9%

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

4. Simplified 76.9%

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

   1. associate-*r* 76.9%

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

   2. fma-neg 76.9%

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

6. Applied egg-rr 76.9%

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

7. Taylor expanded in x around 0 46.5%

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

   1. *-commutative 46.5%

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

   2. metadata-eval 46.5%

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

   3. metadata-eval 46.5%

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

   4. times-frac 46.6%

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

   5. *-rgt-identity 46.6%

      \[\leadsto \frac{\color{blue}{a}}{b \cdot \left(-3\right)} \]

   6. metadata-eval 46.6%

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

9. Simplified 46.6%

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

10. Final simplification 46.6%

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

Developer target: 74.8% 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 2023181 
(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))))