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

Percentage Accurate: 70.4% → 76.8%

Time: 21.4s

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: 70.4% 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: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (cos(y) * (sqrt(x) * 2.0)) - (a / (3.0 * 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 = (cos(y) * (sqrt(x) * 2.0d0)) - (a / (3.0d0 * b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(a / N[(3.0 * 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. *-commutative 72.0%

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

   2. *-commutative 72.0%

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

   3. *-commutative 72.0%

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

   4. *-commutative 72.0%

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

   5. associate-/l* 72.2%

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

   6. *-commutative 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in z around 0 77.9%

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

   1. associate-*r* 77.9%

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

   2. *-commutative 77.9%

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

   3. *-commutative 77.9%

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

7. Simplified 77.9%

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

Alternative 2: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{x} \cdot 2\\ t_2 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-43} \lor \neg \left(t\_2 \leq 10^{-93}\right):\\ \;\;\;\;t\_1 - t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \cos \left(y + z \cdot \left(-0.3333333333333333 \cdot t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (sqrt x) 2.0)) (t_2 (/ a (* 3.0 b))))
   (if (or (<= t_2 -2e-43) (not (<= t_2 1e-93)))
     (- t_1 t_2)
     (* t_1 (cos (+ y (* z (* -0.3333333333333333 t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = sqrt(x) * 2.0;
	double t_2 = a / (3.0 * b);
	double tmp;
	if ((t_2 <= -2e-43) || !(t_2 <= 1e-93)) {
		tmp = t_1 - t_2;
	} else {
		tmp = t_1 * cos((y + (z * (-0.3333333333333333 * t))));
	}
	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) :: tmp
    t_1 = sqrt(x) * 2.0d0
    t_2 = a / (3.0d0 * b)
    if ((t_2 <= (-2d-43)) .or. (.not. (t_2 <= 1d-93))) then
        tmp = t_1 - t_2
    else
        tmp = t_1 * cos((y + (z * ((-0.3333333333333333d0) * t))))
    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 = Math.sqrt(x) * 2.0;
	double t_2 = a / (3.0 * b);
	double tmp;
	if ((t_2 <= -2e-43) || !(t_2 <= 1e-93)) {
		tmp = t_1 - t_2;
	} else {
		tmp = t_1 * Math.cos((y + (z * (-0.3333333333333333 * t))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.sqrt(x) * 2.0
	t_2 = a / (3.0 * b)
	tmp = 0
	if (t_2 <= -2e-43) or not (t_2 <= 1e-93):
		tmp = t_1 - t_2
	else:
		tmp = t_1 * math.cos((y + (z * (-0.3333333333333333 * t))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(sqrt(x) * 2.0)
	t_2 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if ((t_2 <= -2e-43) || !(t_2 <= 1e-93))
		tmp = Float64(t_1 - t_2);
	else
		tmp = Float64(t_1 * cos(Float64(y + Float64(z * Float64(-0.3333333333333333 * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = sqrt(x) * 2.0;
	t_2 = a / (3.0 * b);
	tmp = 0.0;
	if ((t_2 <= -2e-43) || ~((t_2 <= 1e-93)))
		tmp = t_1 - t_2;
	else
		tmp = t_1 * cos((y + (z * (-0.3333333333333333 * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -2e-43], N[Not[LessEqual[t$95$2, 1e-93]], $MachinePrecision]], N[(t$95$1 - t$95$2), $MachinePrecision], N[(t$95$1 * N[Cos[N[(y + N[(z * N[(-0.3333333333333333 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -2.00000000000000015e-43 or 9.999999999999999e-94 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. *-commutative 81.4%

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

      3. *-commutative 81.4%

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

      4. *-commutative 81.4%

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

      5. associate-/l* 81.3%

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

      6. *-commutative 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in z around 0 89.1%

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

      1. associate-*r* 89.1%

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

      2. *-commutative 89.1%

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

      3. *-commutative 89.1%

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

   7. Simplified 89.1%

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

   8. Taylor expanded in y around 0 86.3%

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

   if -2.00000000000000015e-43 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 9.999999999999999e-94

   1. Initial program 56.2%

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

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), a \cdot \frac{-0.3333333333333333}{b}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(0.3333333333333333 \cdot \frac{a}{b \cdot x} + 2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos \left(y + -0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 0.0%

         \[\leadsto \color{blue}{-x \cdot \left(0.3333333333333333 \cdot \frac{a}{b \cdot x} + 2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos \left(y + -0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]

      2. distribute-rgt-neg-in 0.0%

         \[\leadsto \color{blue}{x \cdot \left(-\left(0.3333333333333333 \cdot \frac{a}{b \cdot x} + 2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos \left(y + -0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)\right)} \]

      3. +-commutative 0.0%

         \[\leadsto x \cdot \left(-\color{blue}{\left(2 \cdot \left(\sqrt{\frac{1}{x}} \cdot \left(\cos \left(y + -0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + 0.3333333333333333 \cdot \frac{a}{b \cdot x}\right)}\right) \]

      4. fma-define 0.0%

         \[\leadsto x \cdot \left(-\color{blue}{\mathsf{fma}\left(2, \sqrt{\frac{1}{x}} \cdot \left(\cos \left(y + -0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right), 0.3333333333333333 \cdot \frac{a}{b \cdot x}\right)}\right) \]

   7. Simplified 56.1%

      \[\leadsto \color{blue}{x \cdot \left(-\mathsf{fma}\left(2, \sqrt{\frac{1}{x}} \cdot \left(-1 \cdot \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 44.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(-\mathsf{fma}\left(2, \sqrt{\frac{1}{x}} \cdot \left(-1 \cdot \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)\right)} \]

      2. expm1-undefine 22.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(-\mathsf{fma}\left(2, \sqrt{\frac{1}{x}} \cdot \left(-1 \cdot \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)} - 1} \]

      3. inv-pow 22.9%

         \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(-\mathsf{fma}\left(2, \sqrt{\color{blue}{{x}^{-1}}} \cdot \left(-1 \cdot \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)} - 1 \]

      4. sqrt-pow1 22.9%

         \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(-\mathsf{fma}\left(2, \color{blue}{{x}^{\left(\frac{-1}{2}\right)}} \cdot \left(-1 \cdot \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)} - 1 \]

      5. metadata-eval 22.9%

         \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(-\mathsf{fma}\left(2, {x}^{\color{blue}{-0.5}} \cdot \left(-1 \cdot \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)} - 1 \]

      6. mul-1-neg 22.9%

         \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(-\mathsf{fma}\left(2, {x}^{-0.5} \cdot \color{blue}{\left(-\cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right)}, \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)} - 1 \]

   9. Applied egg-rr 22.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(-\mathsf{fma}\left(2, {x}^{-0.5} \cdot \left(-\cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)} - 1} \]
   10. Step-by-step derivation

       1. sub-neg 22.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(-\mathsf{fma}\left(2, {x}^{-0.5} \cdot \left(-\cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)} + \left(-1\right)} \]

       2. metadata-eval 22.9%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(-\mathsf{fma}\left(2, {x}^{-0.5} \cdot \left(-\cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)} + \color{blue}{-1} \]

       3. +-commutative 22.9%

          \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(x \cdot \left(-\mathsf{fma}\left(2, {x}^{-0.5} \cdot \left(-\cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)}} \]

       4. log1p-undefine 22.9%

          \[\leadsto -1 + e^{\color{blue}{\log \left(1 + x \cdot \left(-\mathsf{fma}\left(2, {x}^{-0.5} \cdot \left(-\cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)}} \]

       5. rem-exp-log 34.4%

          \[\leadsto -1 + \color{blue}{\left(1 + x \cdot \left(-\mathsf{fma}\left(2, {x}^{-0.5} \cdot \left(-\cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)\right)} \]

       6. distribute-rgt-neg-in 34.4%

          \[\leadsto -1 + \left(1 + \color{blue}{\left(-x \cdot \mathsf{fma}\left(2, {x}^{-0.5} \cdot \left(-\cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)}\right) \]

       7. unsub-neg 34.4%

          \[\leadsto -1 + \color{blue}{\left(1 - x \cdot \mathsf{fma}\left(2, {x}^{-0.5} \cdot \left(-\cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right), \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)\right)} \]

       8. fma-define 34.4%

          \[\leadsto -1 + \left(1 - x \cdot \color{blue}{\left(2 \cdot \left({x}^{-0.5} \cdot \left(-\cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right)\right) + \frac{a}{b} \cdot \frac{0.3333333333333333}{x}\right)}\right) \]

   11. Simplified 34.3%

       \[\leadsto \color{blue}{-1 + \left(1 - x \cdot \left(a \cdot \frac{\frac{0.3333333333333333}{x}}{b} - \left({x}^{-0.5} \cdot \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\right) \cdot 2\right)\right)} \]

   12. Taylor expanded in x around inf 54.1%

       \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y + -0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 54.1%

          \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y + -0.3333333333333333 \cdot \left(t \cdot z\right)\right)} \]

       2. associate-*r* 54.5%

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y + \color{blue}{\left(-0.3333333333333333 \cdot t\right) \cdot z}\right) \]

       3. *-commutative 54.5%

          \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y + \color{blue}{z \cdot \left(-0.3333333333333333 \cdot t\right)}\right) \]

   14. Simplified 54.5%

       \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y + z \cdot \left(-0.3333333333333333 \cdot t\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-43} \lor \neg \left(\frac{a}{3 \cdot b} \leq 10^{-93}\right):\\ \;\;\;\;\sqrt{x} \cdot 2 - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y + z \cdot \left(-0.3333333333333333 \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-43} \lor \neg \left(t\_1 \leq 10^{-93}\right):\\ \;\;\;\;\sqrt{x} \cdot 2 - t\_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y + -0.3333333333333333 \cdot \left(z \cdot t\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (or (<= t_1 -2e-43) (not (<= t_1 1e-93)))
     (- (* (sqrt x) 2.0) t_1)
     (* 2.0 (* (sqrt x) (cos (+ y (* -0.3333333333333333 (* z t)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double tmp;
	if ((t_1 <= -2e-43) || !(t_1 <= 1e-93)) {
		tmp = (sqrt(x) * 2.0) - t_1;
	} else {
		tmp = 2.0 * (sqrt(x) * cos((y + (-0.3333333333333333 * (z * t)))));
	}
	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 / (3.0d0 * b)
    if ((t_1 <= (-2d-43)) .or. (.not. (t_1 <= 1d-93))) then
        tmp = (sqrt(x) * 2.0d0) - t_1
    else
        tmp = 2.0d0 * (sqrt(x) * cos((y + ((-0.3333333333333333d0) * (z * t)))))
    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 / (3.0 * b);
	double tmp;
	if ((t_1 <= -2e-43) || !(t_1 <= 1e-93)) {
		tmp = (Math.sqrt(x) * 2.0) - t_1;
	} else {
		tmp = 2.0 * (Math.sqrt(x) * Math.cos((y + (-0.3333333333333333 * (z * t)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if (t_1 <= -2e-43) or not (t_1 <= 1e-93):
		tmp = (math.sqrt(x) * 2.0) - t_1
	else:
		tmp = 2.0 * (math.sqrt(x) * math.cos((y + (-0.3333333333333333 * (z * t)))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if ((t_1 <= -2e-43) || !(t_1 <= 1e-93))
		tmp = Float64(Float64(sqrt(x) * 2.0) - t_1);
	else
		tmp = Float64(2.0 * Float64(sqrt(x) * cos(Float64(y + Float64(-0.3333333333333333 * Float64(z * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	tmp = 0.0;
	if ((t_1 <= -2e-43) || ~((t_1 <= 1e-93)))
		tmp = (sqrt(x) * 2.0) - t_1;
	else
		tmp = 2.0 * (sqrt(x) * cos((y + (-0.3333333333333333 * (z * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-43], N[Not[LessEqual[t$95$1, 1e-93]], $MachinePrecision]], N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] - t$95$1), $MachinePrecision], N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[N[(y + N[(-0.3333333333333333 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -2.00000000000000015e-43 or 9.999999999999999e-94 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. *-commutative 81.4%

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

      3. *-commutative 81.4%

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

      4. *-commutative 81.4%

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

      5. associate-/l* 81.3%

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

      6. *-commutative 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in z around 0 89.1%

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

      1. associate-*r* 89.1%

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

      2. *-commutative 89.1%

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

      3. *-commutative 89.1%

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

   7. Simplified 89.1%

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

   8. Taylor expanded in y around 0 86.3%

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

   if -2.00000000000000015e-43 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 9.999999999999999e-94

   1. Initial program 56.2%

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

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), a \cdot \frac{-0.3333333333333333}{b}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 54.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y + -0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-43} \lor \neg \left(\frac{a}{3 \cdot b} \leq 10^{-93}\right):\\ \;\;\;\;\sqrt{x} \cdot 2 - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y + -0.3333333333333333 \cdot \left(z \cdot t\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (sqrt(x) * 2.0) - (a / (3.0 * 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 = (sqrt(x) * 2.0d0) - (a / (3.0d0 * b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] - N[(a / N[(3.0 * 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. *-commutative 72.0%

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

   2. *-commutative 72.0%

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

   3. *-commutative 72.0%

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

   4. *-commutative 72.0%

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

   5. associate-/l* 72.2%

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

   6. *-commutative 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in z around 0 77.9%

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

   1. associate-*r* 77.9%

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

   2. *-commutative 77.9%

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

   3. *-commutative 77.9%

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

7. Simplified 77.9%

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

8. Taylor expanded in y around 0 68.1%

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

9. Final simplification 68.1%

   \[\leadsto \sqrt{x} \cdot 2 - \frac{a}{3 \cdot b} \]
10. Add Preprocessing

Alternative 5: 65.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] + N[(a * N[(-0.3333333333333333 / 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. *-commutative 72.0%

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

   2. *-commutative 72.0%

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

   3. *-commutative 72.0%

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

   4. *-commutative 72.0%

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

   5. associate-/l* 72.2%

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

   6. *-commutative 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in z around 0 77.9%

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

   1. associate-*r* 77.9%

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

   2. *-commutative 77.9%

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

   3. *-commutative 77.9%

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

7. Simplified 77.9%

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

   1. add-sqr-sqrt 77.8%

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

   2. pow2 77.8%

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

   3. pow1/2 77.8%

      \[\leadsto \cos y \cdot \left({\left(\sqrt{\color{blue}{{x}^{0.5}}}\right)}^{2} \cdot 2\right) - \frac{a}{3 \cdot b} \]

   4. sqrt-pow1 77.8%

      \[\leadsto \cos y \cdot \left({\color{blue}{\left({x}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \cdot 2\right) - \frac{a}{3 \cdot b} \]

   5. metadata-eval 77.8%

      \[\leadsto \cos y \cdot \left({\left({x}^{\color{blue}{0.25}}\right)}^{2} \cdot 2\right) - \frac{a}{3 \cdot b} \]

9. Applied egg-rr 77.8%

   \[\leadsto \cos y \cdot \left(\color{blue}{{\left({x}^{0.25}\right)}^{2}} \cdot 2\right) - \frac{a}{3 \cdot b} \]

10. Taylor expanded in y around 0 68.0%

    \[\leadsto \color{blue}{2 \cdot \sqrt{x} - 0.3333333333333333 \cdot \frac{a}{b}} \]
11. Step-by-step derivation

    1. cancel-sign-sub-inv 68.0%

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

    2. metadata-eval 68.0%

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

    3. associate-*r/ 68.0%

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

    4. associate-*l/ 68.0%

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

    5. *-commutative 68.0%

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

12. Simplified 68.0%

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

13. Final simplification 68.0%

    \[\leadsto \sqrt{x} \cdot 2 + a \cdot \frac{-0.3333333333333333}{b} \]
14. Add Preprocessing

Alternative 6: 50.5% 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%

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

3. Simplified 71.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), a \cdot \frac{-0.3333333333333333}{b}\right)} \]
4. Add Preprocessing

5. Taylor expanded in a around inf 52.5%

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

   1. metadata-eval 52.5%

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

   2. times-frac 52.6%

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

   3. *-lft-identity 52.6%

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

   4. *-commutative 52.6%

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

7. Simplified 52.6%

   \[\leadsto \color{blue}{\frac{a}{b \cdot -3}} \]
8. Add Preprocessing

Alternative 7: 50.4% accurate, 43.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return -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 = (-0.3333333333333333d0) * (a / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

3. Simplified 71.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), a \cdot \frac{-0.3333333333333333}{b}\right)} \]
4. Add Preprocessing

5. Taylor expanded in a around inf 52.5%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{a}{b}} \]
6. Add Preprocessing

Developer Target 1: 74.4% 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 2024123 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -1379333748723514100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 3333333333333333/10000000000000000 z) t)))) (/ (/ a 3) b)) (if (< z 35162906135559870000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 3333333333333333/10000000000000000 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3)))))

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